Random Fields and Stochastic Lagrangian Models: Analysis and Applications in Turbulence and Porous Media 9783110296815, 9783110296648

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Karl K. Sabelfeld Random Fields and Stochastic Lagrangian Models

Karl K. Sabelfeld

Random Fields and Stochastic Lagrangian Models Analysis and Applications in Turbulence and Porous Media

De Gruyter

Physics and Astronomy Classification Scheme 2010: 92.60.hk, AMS 65c05, 76f55.

ISBN 978-3-11-029664-8 e-ISBN 978-3-11-029681-5 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.dnb.de.

© 2013 Walter de Gruyter GmbH, Berlin/Boston

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Preface

This book presents advanced stochastic models and simulation methods for random flows and transport of particles by turbulent velocity fields and flows in porous media. Two main classes of models are constructed: (1) turbulent flows are modeled as synthetic random fields which have certain statistics and features mimicking those of turbulent fluid in the regime of interest, and (2) the models are constructed in the form of stochastic differential equations for stochastic Lagrangian trajectories of particles carried by turbulent flows. In both these classes, we develop Random flight models for the trajectories of tracer particles in turbulence and in flows through porous media. The boundary value problems in stochastic formulation for high-dimensional PDEs present a powerful research instrument in many modern branches of science and technology, in particular, in the turbulence simulation, transport in porous media, random load analysis in mechanical systems, geodesy, composite materials, elastography for biological tissues, acoustic scattering from rough surfaces, defects in metals, X-ray diffraction analysis of epitaxial layers, dislocations in crystals, etc. Interesting example is related to the coagulation of particles carried in turbulent flows governed by Smoluchowski nonlinear systems of coagulation equations with random coefficients. We present detailed results of numerical simulations for these applied problems and discuss stochastic interpretations related to the physics of the relevant problems. A considerable amount of material is devoted to the random field description and different stochastic simulation methods, mainly the stochastic spectral and Fourierwavelet methods for homogeneous vector Gaussian random fields, and the Karhunen– Loève expansions for inhomogeneous random fields. Of special interest are the socalled partially homogeneous random fields which we use in the development of stochastic models for boundary value problems with random boundary conditions, in particular, for Stokes flows, which are presented in the last chapter. The book is written for mathematicians, physicists, and engineers studying processes associated with probabilistic interpretation, researchers in applied and computational mathematics, in environmental and engineering sciences dealing with turbulent transport and flows in porous media, as well as nucleation, coagulation, and chemical reaction analysis under fluctuation conditions. It can be of interest for students and post-graduates studying numerical methods for solving stochastic boundary value problems of mathematical physics and dispersion of particles by turbulent flows and flows in porous media. Acknowledgments. Support of the Russian Fund of Basic Research under grant 1201-00635-à is kindly acknowledged. Berlin, Novosibirsk, June 2012

Karl K. Sabelfeld

Contents

Preface

v

1

Introduction

1

1.1 Why random fields? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3 Fundamental concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Random functions in a broad sense . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Gaussian random vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Gaussian random functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Random fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.5 Stochastic measures and integrals . . . . . . . . . . . . . . . . . . . . . . . 1.3.6 Integral representation of random functions . . . . . . . . . . . . . . 1.3.7 Random trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.8 Stochastic differential, Ito integrals . . . . . . . . . . . . . . . . . . . . . 1.3.9 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.10 Multidimensional diffusion and Fokker–Planck equation . . . . 1.3.11 Central limit theorem and convergence of a Poisson process to a Gaussian process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 9 13 14 16 17 19 21 22 22 25

Stochastic simulation of vector Gaussian random fields

29

2

26

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2 Discrete expansions related to the spectral representations of Gaussian random fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Spectral representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Series expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Expansion with an even complex orthonormal system . . . . . . . 2.2.4 Expansion with a real orthonormal system. . . . . . . . . . . . . . . . 2.2.5 Complex valued orthogonal expansions . . . . . . . . . . . . . . . . . .

30 30 31 31 32 33

2.3 Wavelet expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3.1 Fourier wavelet expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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2.3.2

Wavelet expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3.3

Moving averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4 Randomized spectral models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4.1

Randomized spectral models defined through stochastic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4.2

Stratified RSM for homogeneous random fields . . . . . . . . . . . 39

2.5 Fourier wavelet models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.5.1

Meyer wavelet functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.5.2

Evaluation of the coefficients Fm and Fm

2.5.3

Cut-off parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.5.4

Choice of parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

./

. /

. . . . . . . . . . . . . 40

2.6 Fourier wavelet models of homogeneous random fields based on randomization of plane wave decomposition . . . . . . . . . . . . . . . . . . . . 47 2.6.1

Plane wave decomposition of homogeneous random fields . . . 47

2.6.2

Decomposition with fixed nodes . . . . . . . . . . . . . . . . . . . . . . . . 50

2.6.3

Decomposition with randomly distributed nodes . . . . . . . . . . . 52

2.6.4

Some examples

2.6.5

Flow in a porous media in the first order approximation . . . . . 56

2.6.6

Fourier wavelet models of Gaussian random fields . . . . . . . . . 57

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.7 Comparison of Fourier wavelet and randomized spectral models . . . . . 58 2.7.1

Some technical details of RSM . . . . . . . . . . . . . . . . . . . . . . . . 58

2.7.2

Some technical details of FWM . . . . . . . . . . . . . . . . . . . . . . . . 60

2.7.3

Ensemble averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.7.4

Space averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.9 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3

2.9.1

Appendix A. Positive definiteness of the matrix B . . . . . . . . . 65

2.9.2

Appendix B. Proof of Proposition 2.1 . . . . . . . . . . . . . . . . . . . . 65

Stochastic Lagrangian models of turbulent flows: Relative dispersion of a pair of fluid particles 70 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.2 Criticism of 2-particle models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Contents

3.3 The quasi-1-dimensional Lagrangian model of relative dispersion . . . . 3.3.1 Quasi-1-dimensional analog of formula (2.14a) . . . . . . . . . . . . 3.3.2 Models with a finite-order consistency . . . . . . . . . . . . . . . . . . . 3.3.3 Explicit form of the model (3.26, 3.27) . . . . . . . . . . . . . . . . . . . 3.3.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix 77 78 80 83 88

3.4 A 3-dimensional model of relative dispersion . . . . . . . . . . . . . . . . . . . . 90 3.5 Lagrangian models consistent with the Eulerian statistics . . . . . . . . . . 3.5.1 Diffusion approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Relation to the well-mixed condition . . . . . . . . . . . . . . . . . . . . 3.5.3 A choice of the coefficients ai and bij . . . . . . . . . . . . . . . . . . .

92 92 94 95

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4

A new Lagrangian model of 2-particle relative turbulent dispersion

98

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.2 An examination of Durbin’s nonlinear model . . . . . . . . . . . . . . . . . . . . 98 4.3 Mathematical formulation of a new model . . . . . . . . . . . . . . . . . . . . . . 100 4.4 A qualitative analysis of the problem (4.14) for symmetric ./ . . . . . 102 4.4.1 Analysis of the problem (4.14) in the deterministic case . . . . . 102 4.4.2 Analysis of the problem (4.14) for stochastic ./ . . . . . . . . . . 103 4.5 Qualitative analysis of the problem (4.14) in the general case . . . . . . . 108 5

The combined Eulerian–Lagrangian model

113

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.2 2-particle models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.2.1 Eulerian stochastic models of high-Reynolds-number pseudoturbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.3 A new 2-particle Eulerian–Lagrangian stochastic model . . . . . . . . . . . 120 5.3.1 Formulation of 2-particle Eulerian–Lagrangian model . . . . . . . 120 5.3.2 Models for the p. d. f. of the Eulerian relative velocity . . . . . . . 123 5.4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6

Stochastic Lagrangian models for 2-particle relative dispersion in high-Reynolds-number turbulence

129

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

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6.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.3 A closure of the quasi-1-dimensional model of relative dispersion . . . 131 6.4 Choice of the model .6.1/ for isotropic turbulence . . . . . . . . . . . . . . . . 132 6.5 The model of relative dispersion of two particles in a locally isotropic turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.5.1 Specification of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.5.2 Numerical analysis of the Q1D-model .6.30/ . . . . . . . . . . . . . . 137 6.6 Model of the relative dispersion in intermittent locally isotropic turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7

Stochastic Lagrangian models for 2-particle motion in turbulent flows. Numerical results 142 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.2 Classical pseudoturbulence model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.2.1 Randomized model of classical pseudoturbulence . . . . . . . . . . 143 7.2.2 Mean square separation of two particles in classical pseudoturbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 7.3 Calculations by the combined Eulerian–Lagrangian stochastic model 7.3.1 Mean square separation of two particles . . . . . . . . . . . . . . . . . . 7.3.2 Thomson’s “two-to-one” reduction principle . . . . . . . . . . . . . . 7.3.3 Concentration fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

149 149 152 154

7.4 Technical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 8

The 1-particle stochastic Lagrangian model for turbulent dispersion in horizontally homogeneous turbulence 159 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.2 Choice of the coefficients in the Ito equation . . . . . . . . . . . . . . . . . . . . 162 8.3 2D stochastic model with Gaussian p. d. f. . . . . . . . . . . . . . . . . . . . . . . 164 8.4 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

9

Direct and adjoint Monte Carlo for the footprint problem

171

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

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9.2 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 9.3 Stochastic Lagrangian algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 9.3.1

Direct Monte Carlo algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 174

9.3.2

Adjoint algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

9.4 Impenetrable boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 9.5 Reacting species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 9.6 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 9.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 9.8 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 9.8.1

Appendix A. Flux representation . . . . . . . . . . . . . . . . . . . . . . . 188

9.8.2

Appendix B. Probabilistic representation . . . . . . . . . . . . . . . . . 188

9.8.3

Appendix C. Forward and backward trajectory estimators . . . . 189

10 Lagrangian stochastic models for turbulent dispersion in an atmospheric boundary layer

193

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 10.2 Neutrally stratified boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 10.2.1 General case of Eulerian p. d. f. . . . . . . . . . . . . . . . . . . . . . . . . . 197 10.2.2 Gaussian p. d. f. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 10.3 Comparison with other models and measurements . . . . . . . . . . . . . . . 201 10.3.1 Comparison with measurements in an ideally-neutral surface layer (INSL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 10.3.2 Comparison with the wind tunnel experiment by Raupach and Legg (1983) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 10.4 Convective case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 10.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 10.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 10.7 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 10.7.1 Appendix A. Derivation of the coefficients in the Gaussian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 10.7.2 Appendix B. Relation to other models . . . . . . . . . . . . . . . . . . . 215

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11 Analysis of the relative dispersion of two particles by Lagrangian stochastic models and DNS methods

218

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 11.2 Basic assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 11.2.1 Markov assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 11.2.2 Consistency with the second Kolmogorov similarity hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 11.2.3 Thomson’s well-mixed condition . . . . . . . . . . . . . . . . . . . . . . . 222 11.3 Well-mixed Lagrangian stochastic models . . . . . . . . . . . . . . . . . . . . . . 222 11.3.1 Quadratic-form models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 11.3.2 Quasi-1-dimensional models . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 11.3.3 3-dimensional extension of Q1D models . . . . . . . . . . . . . . . . . 225 11.4 Stochastic Lagrangian models based on the moments approximation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 11.4.1 Moments approximation conditions . . . . . . . . . . . . . . . . . . . . . 226 11.4.2 Realizability of LS models based on the moments approximation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 11.5 Comparison of different models of relative dispersion for the inertial subrange of a fully developed turbulence . . . . . . . . . . . . . . . . . . . . . . . 229 11.5.1 Q1D quadratic-form model of Borgas and Yeung . . . . . . . . . . 229 11.5.2 Comparison of different models in the inertial subrange . . . . . 231 11.6 Comparison of different Q1D models of relative dispersion for modestly large Reynolds number turbulence (Re ' 240) . . . . . . . . . 232 11.6.1 Parametrization of Eulerian statistics . . . . . . . . . . . . . . . . . . . . 232 11.6.2 Bi-Gaussian p. d. f. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 11.6.3 Q1D quadratic-form model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 12 Evaluation of mean concentration and fluxes in turbulent flows by Lagrangian stochastic models

238

12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 12.2 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 12.3 Monte Carlo estimators for the mean concentration and fluxes . . . . . . 243 12.3.1 Forward estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

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12.3.2 Modified forward estimators in case of horizontally homogeneous turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 12.3.3 Backward estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 12.4 Application to the footprint problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 12.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 12.6 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.1 Appendix A. Representation of concentration in Lagrangian description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.2 Appendix B. Relation between forward and backward transition density functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.3 Appendix C. Derivation of the relation between the forward and backward densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Stochastic Lagrangian footprint calculations over a surface with an abrupt change of roughness height

253 253 255 255 258

13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 13.2 The governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 13.2.1 Evaluation of footprint functions . . . . . . . . . . . . . . . . . . . . . . . 260 13.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 13.3.1 Footprint functions of concentration and flux . . . . . . . . . . . . . . 263 13.4 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 13.5 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 13.5.1 Appendix A. Dimensionless mean-flow equations . . . . . . . . . . 277 13.5.2 Appendix B. Lagrangian stochastic trajectory model . . . . . . . . 278 14 Stochastic flow simulation in 3D porous media

280

14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 14.2 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 14.3 Direct numerical simulation method: DSM-SOR . . . . . . . . . . . . . . . . . 284 14.4 Randomized spectral model (RSM) . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 14.5 Testing the simulation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 14.6 Evaluation of Eulerian and Lagrangian statistical characteristics by the DNS-SOR method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 14.6.1 Eulerian statistical characteristics . . . . . . . . . . . . . . . . . . . . . . . 292

xiv

Contents

14.6.2 Lagrangian statistical characteristics . . . . . . . . . . . . . . . . . . . . . 294 14.7 Conclusions and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 15 A Lagrangian stochastic model for the transport in statistically homogeneous porous media

300

15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 15.2 Direct simulation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 Random flow model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.2 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.3 Evaluation of Eulerian characteristics . . . . . . . . . . . . . . . . . . . . 15.2.4 Evaluation of Lagrangian characteristics . . . . . . . . . . . . . . . . .

301 301 303 306 310

15.3 Construction of the Langevin-type model . . . . . . . . . . . . . . . . . . . . . . . 15.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.2 Langevin model for an isotropic porous medium . . . . . . . . . . . 15.3.3 Expressions of the drift terms . . . . . . . . . . . . . . . . . . . . . . . . . .

314 314 316 319

15.4 Numerical results and comparison against the DSM . . . . . . . . . . . . . . . 321 15.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 16 Coagulation of aerosol particles in intermittent turbulent flows

326

16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 16.2 Analysis of the fluctuations in the size spectrum . . . . . . . . . . . . . . . . . . 329 16.3 Models of the energy dissipation rate . . . . . . . . . . . . . . . . . . . . . . . . . . 332 16.3.1 The model by Pope and Chen (P&Ch) . . . . . . . . . . . . . . . . . . . 332 16.3.2 The model by Borgas and Sawford (B&S) . . . . . . . . . . . . . . . . 334 16.4 Monte Carlo simulation for the Smoluchowski equation in a stochastic coagulation regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.1 The total number of clusters and the mean cluster size . . . . . . . 16.4.2 The functions N3 .t / and N10 .t / . . . . . . . . . . . . . . . . . . . . . . . . 16.4.3 The size spectrum Nl for different time instances . . . . . . . . . . 16.4.4 Comparative analysis for two different models of the energy dissipation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

335 337 339 340 341

16.5 The case of a coagulation coefficient with no dependence on the cluster size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 16.6 Simulation of coagulation processes in turbulent coagulation regime

343

Contents

xv

16.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 16.8 Appendix. Derivation of the coagulation coefficient . . . . . . . . . . . . . . . 346 17 Stokes flows under random boundary velocity excitations

349

17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 17.2 Exterior Stokes problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 17.2.1 Poisson formula in polar coordinates . . . . . . . . . . . . . . . . . . . . 353 17.3 K-L expansion of velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 17.3.1 White noise excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 17.3.2 General case of homogeneous excitations . . . . . . . . . . . . . . . . 361 17.4 Correlation function of the pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.1 White noise excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.2 Homogeneous random boundary excitations . . . . . . . . . . . . . . 17.4.3 Vorticity and stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

366 366 368 368

17.5 Interior Stokes problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 17.6 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 Bibliography

381

Index

397

Chapter 1

Introduction

1.1

Why random fields?

Probabilistic approach and stochastic simulation are becoming more and more popular in all branches of science and technology, especially in problems where the data are randomly fluctuating, or they are highly irregular in a deterministic sense. As a rule, in such problems it is very difficult and expensive to carry out measurements to extract the desired data. Important examples we deal with in this book are the turbulent flow simulation [146], and construction of flows through porous media [34, 65]. The temporal and spatial scales of the input parameters in this class of problems vary enormously, and their behavior is very complicated, so that there is no chance of describing it deterministically. In the stochastic approach, the input parameters are considered as random fields, and one needs to know only a few of their functions, such as the mean and correlation tensor, whose behavior in time and space is much more regular, so that usually it is considerably easier to extract them through measurements. In most applications, it is assumed that the random fields are Gaussian, or that they can be obtained by a functional transformation of Gaussian fields. Generally, it is very difficult to construct efficient simulation methods for inhomogeneous random fields, even if they are Gaussian. Therefore, the most developed methods deal with homogeneous or quasihomogeneous random fields where the characteristic scales of the variations of the means of the field are considerably larger than the correlation scale. There are highly intensive studies and literature concerned with the simulation of homogeneous random fields. The most important class of simulation methods seeks to construct stochastic models based on spectral representations. We shall consider general real-valued Gaussian homogenous random fields u.x/ defined on the multidimensional Euclidean space IRd . Under quite general conditions, a real-valued Gaussian homogenous random field u.x/ can be represented through a stochastic Fourier integral [146]: Z u.x/ D

IRd

e 2ikx E 1=2 .k/WQ .dk/,

(1.1)

where WQ .dk/ is a complex-valued white noise random measure on IRd , with WQ .B/ D WQ .B/, hWQ .B/i D 0, and hWQ .B/WQ .B 0 /i D .B \ B 0 / for the Lebesgue measure  and all Lebesgue-measurable sets B, B 0 . We use angle brackets hi to denote statistical (ensemble) averages. The spectral density E.k/ is a nonnegative even function

2

Chapter 1 Introduction

representing the strength (energy) of the random field associated to the wavenumber k, meaning the length scale 1=jkj and direction k=jkj. We mention several simulation methods based on the spectral representation: (i) the discrete spectral method (DSM) [226] which is simply a deterministic discrete approximation of the Fourier Stieltjes integral; (ii) the randomized spectral method (RSM) [102, 142, 191]) which is based on a randomized approximation of the same Fourier Stieltjes integral; (iii) the Fourier wavelet method (FWM) [49,50,104,118] is a different approximation of the Fourier Stieltjes integral based on reexpansion in a special family of orthogonal functions, and is obtained by an expansion of the Gaussian white noise in a wavelet basis. Another class of methods includes methods which deal with the expansions in the physical space, in the relevant system of orthonormal functions: (i) methods based on expansions in the wavelet basis (WM) [231, 264]; (ii) the Karhunen–Loève expansion method (K–L) [230, 249] based on the expansions in eigenfunctions of the correlation operator – note that this also works for inhomogeneous random fields; (iii) the moving averages mMethod (MAM) [140], based on the representation of the random field in the form of a convolution of a deterministic function (more precisely, a Fourier transform of a square root of the spectral function) with the Gaussian white noise in the physical space. We mention the the fast Fourier transform spectral method (FFTSM) (e. g., see [39, 168]) which is a particular case of the discrete spectral method whose nodes are chosen as a diadic mesh to apply further the fast Fourier method. The matrix factorization method (MFM) [40, 224] and the circulant embedding method (CEM) [41] are based on the Holessky decomposition of the covariance matrix. The methods listed above all have their advantages as well as their disadvantages. For example, DSM, RSM, and MAM are simple and convenient for implementation; they provide the possibility to calculate the values of the random field at some points on demand. But in multidimensional cases, DSM and MAM are less efficient. FFTSM is also simple for implementation, but it calculates the random field only on a diadic mesh and has therefore a disadvantage that the samples are periodic. Furthermore, FWM and WM models are efficient for simulating multiscale processes but they are difficult in implementation. The K–L model is highly efficient but is not universal, since it is necessary to solve the eigenvalue problem for the correlation operator. More details about the above mentioned methods can be found in [25, 47, 54, 103, 104, 165] where a comparative analysis of some methods is also given. In particular, in [25, 47], RSM and FWM are compared by analyzing a fractal random field with the spectral function F .k/ D k ˛ .1 < ˛ < 3/, where the calculated structure function was compared with the exact result. The main conclusion is that to construct the samples of a multiscale random field with a fixed desired accuracy, the cost of RSM is considerably lower than that of FWM if lg.lmax = lmin /  4 where lmin and lmax are the minimal and maximal spatial scales of the random field, respectively. In [104] we have shown that a logarithmically uniform subdivision of the spectral space (we have

Section 1.2 Some examples

3

introduced such a subdivision in [119]) when calculating two- and a few-point statistical characteristics of the fractal random field, the RSM is more efficient than FWM for all values of lmax = lmin . In particular, when calculating the structure function of a multiscale random field with ˛ D 5=3, lmax = lmin D 1012 it was found that the cost of FWM was 12 times larger than that of RSM; results were obtained for 9 decades, with a fixed accuracy. Up to now, we discussed the calculation of statistical characteristics by ensemble averaging over the samples constructed by the relevant method. In many practical problems (e. g., in underground hydrology) only data obtained through spatial averaging is at hand, for instance, statistical characteristics obtained by spatial averages, or over a family of Lagrangian trajectories generated in one fixed sample of the field (e. g., see [35, 65]). If the random field is ergodic, then the ensemble averages can be well approximated by the appropriate space averages. This is very important when a boundary value problem with random parameters is solved: then, in contrast to the ensemble averaging, we have to solve the problem only once and then make the relevant space averaging. In practical calculations, to increase the efficiency, it is sometimes reasonable to combine both the space and ensemble averaging, e. g., see [99,120]. The same technique is used also in simulation of turbulent transport [63, 240]. And so we stress that good ergodic properties of the constructed random field model are very important and highly desired in practical problems. In [104] we studied the ergodic properties of RSM and FWM. Calculations of structure functions through ensemble and space averaging have shown that the ergodic properties of FWM are much better than those of RSM. Therefore, to obtain a good approximation through space averaging in RSM, it is necessary to take many thousands of harmonics per each decade. However, this conclusion was made only for random processes (i. e., random fields depending on one scalar variable). Thus, as discussed above, random fields provide a useful mathematical framework for representing disordered heterogeneous media in theoretical and computational studies. The random fields may appear in a very simple form in the problem, and then the focus is only on a detailed and accurate simulation of the samples. For instance, in many optic problems there is a need to simulate a random surface with a given correlation function, while the reflected light intensity is then easily calculated as in integral over a certain angle region. In more sophisticated models, random fields enter differential or integral equations in the form of a coefficient, a kernel, a right-hand side, a boundary condition, or even the boundary itself.

1.2

Some examples

Let us give some examples. First we mention turbulent transport, where the velocity field representing the turbulent flow is modeled as a random field vE.x, t / with statistics

4

Chapter 1 Introduction

encoding important empirical features, and the temporal dynamics of the position XE .t / E and velocity VE .t / D ddtX of immersed particles is then governed by equations involving this random field such as p   (1.2) m dVE .t / D  VE .t /  vE.XE .t /, t / d t C 2kB T  d W.t /, where m is particle mass,  is its friction coefficient, kB is Boltzmann’s constant, T is the absolute temperature, and W.t / is a random Wiener process representing molecular collisions. In more complicated stochastic models of turbulent flows, both the drift and the dispersion terms are constructed as some functions using data extracted from physical and statistical laws. Let us consider a passive scalar dispersed by the turbulent velocity field. The passive scalar is assumed to follow the streamlines of the flow. We assume that the source of particles is quite arbitrary; for instance, it might be situated on the surface or in the space, or even at given points. Let us denote by q.x, t / the spatial-temporal density distribution function of the source, i. e„ the number of emitted particles per unit volume in a unit time interval at the phase point .x, t /. Initially, the spatial density of particles is given by q0 .x/. The particles are transported by a 3D turbulent velocity field uE .x, t / D .u1 .x, t /, u2 .x, t /, u3 .x, t //. Let us denote by X.t ; x0 , t0 / and V.t ; x0 , t0 / D uE .X.t ; x0 , t0 /, t / the Lagrangian spatial coordinates and the velocity, respectively. Then, neglecting molecular diffusion, the instantaneous concentration c.x, t / is governed by @c @c.x, t / X ui .x, t / D q.x, t /, C @t @xi 3

t > 0,

c.x, 0/ D q0 .x/.

(1.3)

iD1

The turbulent velocity field uE .x, t / is considered to be an incompressible 3D random field. Accordingly, the concentration c.x, t /, satisfying equation (1.3) with random coefficients ui , is a scalar random field, and we are interested in calculating the mean concentration hc.x, t /i and the mean fluxes hui .x, t /c.x, t /i, i D 1, 2, 3. These functions can be evaluated by tracking Lagrangian trajectories, which are obtained in turn by solving the system of stochastic differential equations d X.t / D V.t /dt , d V.t / D a.t , X.t /, V.t //dt C

p

C0 "N.X.t /, t / d W.t /,

where the function a is to be defined in each specific situation, C0 is the universal Kolmogorov constant (C0  4 + 6), "N.x, t / is the mean dissipation rate of the kinetic energy of turbulence, and W.t / is a standard 3D Wiener process. A second example concerns transport through porous media, such as groundwater aquifers, in which the hydraulic conductivity K.x/ is modeled as a random field reflecting the empirical variability of the porous medium. The Darcy flow rate qE.x/ in

5

Section 1.2 Some examples

response to pressure applied at the boundary is governed by the Darcy equation qE.x/ D K.x/ grad .x/,

(1.4)

div qE D 0, in which the random hydraulic conductivity function appears as a coefficient, and the applied pressure is represented in the boundary conditions for the internal pressure head . Our concern is with the computational simulation of random fields for applications such as these. Interesting insights into the dynamics of transport in disordered media can be achieved already through relatively simple random models for the velocity field, such a finite superposition of Fourier modes, with each amplitude independently evolving according to an Ornstein–Uhlenbeck process. Here efficient and accurate numerical simulations of the flow can be achieved through application of the well-developed literature on simulating stochastic ordinary differential equations. In conventional deterministic numerical methods, boundary value problems for random PDEs are solved as follows. First, one constructs a synthesized sample of the input random parameter. Then the obtained deterministic equation is solved numerically, say, by the finite element method, and gives the solution in all points of the grid domain. These two steps are repeated many times, so that the obtained statistics are sufficient for calculation of the desired sufficiently accurate averages. This approach is used in stochastic finite element methods (e. g., see [2, 150, 230, 249]). Obviously, this technique is generally time consuming, and to solve problems of practical interest one needs supercomputers to extract sufficient statistical information. In the Monte Carlo approach, the algorithms are designed so that the solution is calculated only in the desired set of points without constructing the solution in the whole domain (e. g., see [191, 203, 204]). To evaluate different statistical characteristics of random boundary value problems we use the double randomization technique (e. g., see [191]). This approach is possible if the desired statistical characteristics (e. g., the mean or the correlation tensor) can be represented in the form of a double expectation over the input random parameters, and over the trajectories of a Markov process used in a stochastic estimator for solving the deterministic equation. The advantage of this method is that there is no need to solve the equation many times, hence, the cost of this method is drastically decreased compared to the stochastic finite element method. The well-known drawback of stochastic simulation methods should be mentioned: the error behaves like "  O.N 1=2 /, where N is the number of samples; hence, it is reasonable to apply the Monte Carlo methods if the desired accuracy is not too high. So, for realistic applied problems the typical Monte Carlo accuracy lies in the range of 0.1 % to several percent. The basic idea behind double randomization can be explained by the following very simple example. Assume we have to evaluate an integral Z f .x, y; !/dy, (1.5) J.x; !/ D D

6

Chapter 1 Introduction

where f .x, y; !/ is a random function indexed through x, y, defined on a probability space, ! being the relevant random element. Obviously of interest are statistical characteristics of the random process J.x; !/, like the expectation hJ.x; !/i and the covariance hJ.x1 ; !/J.x2 ; !/i. In deterministic methods, to calculate the expectation hJ.x; !/i, one has first to construct a sample of the random function f , say, f .x, y; !1 /, and then calculate the integral J.x; !1 / by one of the quadrature formulas. This is then repeated N times, N large enough to guarantee that the average over N samples provides a good approximation to hJ.x; !/i. Thus, in short, one must solve a deterministic problem (in this case, evaluation of the integral) N times, N being the number of samples. Double randomization is based on the representation of the desired functional as a double expectation. Indeed, we choose an arbitrary probability density function p.y/, y 2 D, arbitrary enough but so that p.y/ ¤ 0 for y, where f .x, y; !/ ¤ 0 for all x and !. Then we can write   (1.6) J.x; !/ D Ep f .x, ; !/=p./ where Ep stands for the average over the random points  distributed in D according to the density p. Therefore, ˛   ˝  (1.7) hJ.x; !/i D Ep f .x, ; !/=p./ D E.!,p/ f .x, ; !/=p./ , where E.!,/ stands for averaging over random elements .!, /. This statement is exactly the Fubini theorem, and it shows that the desired result can be obtained by averaging over random samples of ! and . This approach works also when the deterministic problem is not simply an integral evaluation, but a PDE, or an integral equation, or nonlinear system of equations, e. g., like the Smoluchowski equations. The main challenging problem here is to transform the solution of the original problem to the evaluation of an expectation over relevant stochastic elements, including averaging in functional spaces. In our case the expectations are often constructed over Markov chains and Gaussian random fields. Note that in applied problems, we deal also with generalized random fields, but in Monte Carlo simulations they do not complicate the situation, but, quite the contrary, the stochastic simulation algorithms can efficiently use this feature. Let us illustrate this by the following simple but important example when the input randomness enters the problem as a generalized 2D random field u.x, !/, x 2 Œ0, L, ! D .!1 , !2 , : : : , !m /, in the form: u.x, !/ D

m X

ı.x  !j /,

(1.8)

j D1

where ı is the Dyrac delta function, and the random points !1 , !2 , : : : , !m are distributed on Œ0, L with a density p.!/ independent of the spatial coordinate x. The

7

Section 1.2 Some examples

random points !1 , !2 , : : : , !m may have a quite different distribution on Œ0, L, i. e., they may be all independent of each other, they may form a Markov chain with a certain transition probability density, or they may be placed almost periodically, with small but correlated random shifts !j from a mean fixed step hı!j i: !j C1 D !j Cı!j where hı!j i D 1 D m=L is the mean density of the points on Œ0, L. Thus the random field u.x, !/ is stationary with the mean hu.x, y; !/i D , as can be found by direct calculation. This kind of random process enters the boundary elastic displacements produced by the nets of misfit dislocations in crystals [87], in many problems related to renewal processes, in the analysis of statistic of neuronal spike trains (for instance, see [52, 68, 180]), etc. In the analysis of dislocations in crystals, the goal is to evaluate the x-ray diffraction peak profiles from distributions of misfit and threading dislocations [87]. The x-ray scattering amplitude from a film of thickness d is given by an integral of the form Z 1 Z d ® ¯ dx dz exp iŒqx x C qz z C V .x, z/ , (1.9) A.qx , qz / D 1

0

Z

where V .x, z/ D

K.x  x 0 , z/u.x 0 , !/dx 0

(1.10)

is the resulting displacement field due to all misfit dislocations, the kernel K.x  0 x 0 , z/ is the Green function given explicitly,˝ and the random ˛ field u.x , !/ is defined by 2 (1.8). The scattered intensity I.qx , qz / D jA.qx , qz /j can be directly calculated by the Monte Carlo double randomization method, as explained above, by a randomized evaluation of the integral representation: Z 1 Z dZ d ˝ ˛ dx dz1 dz2 e iŒqx xCqz .z1 z2 / e iŒV .x1 ,z1 /V .x2 ,z2 / . (1.11) I.qx , qz / D 1

0

0

Note that in the case where the random field u is Gaussian, it is possible to evaluate the expectation explicitly (see [87]). Thus the boundary value problems with random coefficients, parameters, random source terms, stochastically distributed boundary functions, or even with randomly moving boundaries are used as a powerful instrument in modern science and technology. We mention here applied fields such as structural mechanics, composite materials [2], porous media and soils [34, 99, 196, 260], biological tissues [258], geodesy [182, 212], turbulence, [13, 103, 104, 146, 191], etc. In engineering-related stochastic boundary value problems, the common computational techniques include Monte Carlo methods, stochastic finite elements, finite difference, and spectral methods. Among these methods, the finite volume and boundary element techniques are the methods most adaptable to problems in solid and structural mechanics characterized with highly irregular and complex structures [2, 230, 249]. We mention also classical potential

8

Chapter 1 Introduction

problems dealing with random boundary conditions and sources [31] where the Monte Carlo methods are very efficient (e. g., see [191, 206–208]). The book is organized as follows. Chapter 2 presents different simulation methods for Gaussian random fields. Chapter 3 deals with the stochastic Lagrangian models of relative dispersion of a pair of fluid particles. In Chapter 4 a new version of the 2-particle relative turbulent dispersion model is developed. A combined Eulerian– Lagrangian model is presented in Chapter 5. In Chapter 6 we describe a stochastic Lagrangian model extended to the intermittent turbulence. Chapter 7 presents results of numerical experiments. A 1-particle stochastic Lagrangian model for a horizonatlly homogeneous turbulent flow is described in Chapter 8. Formulation of the footprint problem and the methods based on backward Lagrangian trajectories are presented in Chapter 9. Applications of the stochastic Lagrangian models to evaluate the particle transport in the boundary layer of the atmosphere are given in Chapter 10. Comparisons of different 2-particle models are described in Chapter 11. Algorithms for concentration and fluxes in turbulent flows by stochastic Lagrangian models are presented in Chapter 12. Application to the footprint problem for the case of an abrupt change of roughness is given in Chapter 13. A Lagrangian stochastic model for the transport in porous medium is presented in Chapters 14 and 15. Chapter 16 deals with the analysis of the coagulation of aerosol particles in intermittent turbulent flows. Finally, in Chapter 17 we give an example of a Stokes flow which is governed by Stokes equations with random boundary conditions.

1.3 Fundamental concepts We introduce two main classes of stochastic simulation models presented in this book: an Eulerian class which is based on a random velocity field model defined on a fixed coordinate system. The velocity field is generated over a prescribed spatial domain, but by a direct stochastic construction rather than the much more expensive simulation of the nonlinear Navier–Stokes PDE’s. In the Lagrangian class the motion of fluid particles is stochastically modeled as a random trajectory X.t / of any immersed particle representing, for example, a tracer, pollutant, or chemical reactant. It is computed using the local value of the stochastically constructed velocity. The random trajectory thus can be defined as a random D u.t ; X.t / process determined from the random ordinary differential equation dX.t/ dt where u. , r/ is a random field, or, alternatively, it can be determined from a stochastic Ito-type differential equation dX.t / D A dt C B d W .t /. These two classes of models use different mathematical apparatus: the Eulerian stochastic models focus on the simulation of spatial-temporal random fields with a desired statistical characteristics, while the stochastic Lagrangian models are based on the so-called Ito-type stochastic differential equations, known in physics as Langevintype equations.

9

Section 1.3 Fundamental concepts

As mentioned in [103], the primary challenge in most Eulerian fluid Monte Carlo simulations is the generation of a synthetic random velocity u.x; t / which has certain statistics and features mimicking those of a turbulent fluid in the regime of interest. Thus a fully developed turbulent flow at a sufficiently high Reynolds number should possess a wide inertial range of scales over which the statistics of the velocity field assume a self-similar fractal structure. A quantitative way to express this criterion is hju.x C r; t /  u.x; t /j2 i D SvI jrj2H

for LK  jrj  L0 ,

where hi denotes a statistical average, 0 < H < 1 is the Hurst exponent which takes the Kolmogorov value H D 1=3 for fully developed turbulence, LK is the dissipation length scale and L0 is the integral length scale, which define the extent of the inertial scaling range, and SvI is a (dimensional) scaling prefactor. Other desired properties in turbulence simulations are the incompressibility of the fluid and appropriate geometric symmetries such as isotropy.

1.3.1

Random functions in a broad sense

Let us start with an informal description of random functions. A random function, as well as a deterministic function, is defined by some dependence . / which describes a mapping from the space of parameters 2 ‚ to the space of values of . /, thus it can be real- or complex-valued, or generally vector-valued. But in the stochastic case, . / is a random value for each fixed . Thus, a random function is a family of random variables . / D . , !/ defined on a probability space . , F , P / depending on a parameter 2 ‚. The deterministic function . , !0 / for a fixed value ! D !0 is called a sample function (or a sample trajectory). Here we recall that a finite set of random variables 1 , 2 , : : : n is fully defined by the mutual distribution function ¯ ® F .x1 , x2 , : : : , xn / D P 1 < x1 , 2 < x2 , : : : , n < xn .

(1.12)

In the case of random functions, we have to characterize an infinite family of random variables. So we can say that the infinite family of random variables . / is defined if statistical characteristics of any finite sets of random variables . 1 /, . 2 /, : : : . n /,

i 2 ‚,

i D 1, 2 : : : , n;

n D 1, 2 : : : ,

(1.13)

i. e., any finite-dimensional distributions are defined, which means, the random function . / is defined by its finite-dimensional distributions F1 ,2 ,:::,n .x1 , x2 , : : : , xn /,

i 2 ‚;

i D 1, 2 : : : , n;

n D 1, 2 : : : ,

(1.14)

and any function F1 ,2 ,:::,n .x1 , x2 , : : : , xn / is interpreted as a mutual distribution function of the set of random variables (1.13).

10

Chapter 1 Introduction

To make this interpretation correct, the family of distributions should satisfy some assumptions. These assumptions are quite natural: F1 ,2 ,:::,n ,nC1 ,:::,nCm .x1 , x2 , : : : , xn , C1, : : : , C1/ D F1 ,2 ,:::,n .x1 , x2 , : : : , xn /, (1.15) F1 ,2 ,:::,n .x1 , x2 , : : : , xn / D F1 ,2 ,:::,n .xi1 , xi2 , : : : , xin /,

(1.16)

where i1 , i2 , : : : , in is any permutation of the indices 1, 2 : : : , n. Thus we are in a position to give the following definition. Random function on a set of parameters 2 ‚ having real values . / is defined as a family of distributions (1.14) which satisfies the conditions (1.15) and 1.16). The functions F1 ,2 ,:::,n .x1 , x2 , : : : , xn / are called finite-dimensional distributions of the random function . /. This definition is clear and simple, and is sufficient when we are interested in statistical characteristics for a finite set of the parameter’s values. But it is not satisfactory when we are trying to characterize the function in its entirety, for all infinite values of the parameter. For instance, this definition cannot provide us with a graph of the random function. With this definition, we cannot even answer such an important question as whether the sample of the random function is continuous or differentiable. A different definition considers the random function as an element of an appropriate functional space and will be given later. In this section we deal with the definition of the random function in athewise sense given above by the family of distribution functions. Generalization to vector random functions is obvious: a vector random function . / is defined as a vector with n components, scalar random functions: . / D .1 . /, 2 . /, : : : n . //. The distribution function of this vector random function is the function of nm variables F1 ,2 ,:::,n .x11 , x12 , : : : , xnm /

¯ ® D P 1 . 1 / < x11 , 1 . 2 / < x12 , : : : , n . m / < xmn .

The distribution functions can be represented through the probability density f1 ,:::,n .x1 , : : : , xn /: Z x1 Z xn F1 ,:::,n .x1 , : : : , xn / D  f1 ,:::,n .y1 , : : : , yn / dy1 : : : yn . 1

1

From this the following well-known property follows f1 ,:::,n .x1 , : : : , xn / Z x1 Z D  1

xn

1

f1 ,:::,n ,nC1 ,:::nCm .x1 , : : : xn , y1 , : : : , ym / dy1 : : : ym .

11

Section 1.3 Fundamental concepts

A characteristic function of a finite-dimensional distribution is defined by ² X ³ n '1 ,:::,n D E exp i . k /uk , kD1

where E stands for the mathematical expectation, and u1 , : : : un are real numbers. If the density f1 ,:::,n exists, then Z '1 ,:::,n D

n P

e i D1

xi ui

IRn

f1 ,:::,n .x1 , : : : , xn / dx1 : : : dxn ,

i. e., the characteristic function is a Fourier transform of the probability density. Moment functions of . / are defined by mj1 ,:::,js . 1 , : : : , s / D EŒ. 1 /j1 : : : Œ. s /js ,

jk  0,

.k D 1, 2, : : : , s/

if the expectation on the right-hand side exists for all i 2 ‚, i D 1, 2, : : : , s. A random function . / belongs to the class Lp .‚/, . 2 Lp .‚/) if Ej. /jp < 1 for all 2 ‚. So if . 2 Lp .‚/, then all moments of the order q  p are finite. If the characteristic functions of the finite-dimensional distributions are given, then the moments of any integer order can be obtained by taking derivatives. Indeed, if . 2 Lp .‚/, then mj1 ,:::,js . 1 , : : : , s / D .1/q

@q '.u1 , : : : , us / j

j

@u11 : : : us s

for q  p where q D j1 C    C js . Centered moments are defined by  j  j m N j1 ,:::,js . 1 , : : : , s / D E . 1 /  m1 . 1 / 1 : : : . s /  m1 . s / s . Here mi . / is the expectation of i . /: mi . / D Ei . /. The correlation function R. 1 , 2 / is defined by    R. 1 , 2 / D E . 1 /  m. 1 / . 2 /  m. 2 / . The variance is the quantity  2 . / D R. , /, and the correlation coefficient is just the normalized correlation function: r. 1 , 2 / D

R. 1 , 2 / .  . 1 /  . 2 /

12

Chapter 1 Introduction

If the random variables . 1 / and . 2 / are independent, then the correlation coefficient is zero. Note that the reverse is not true: zero correlation does not imply that the variables are independent. However, if the mutual 2D distribution of . 1 / and . 2 / is Gaussian, then the zero correlation implies that . 1 / and . 2 / are independent. Generalization to complex-valued random functions is straightforward. A complexvalued random function is defined by . / D . / C i . / and can be considered as a 2-dimensional vector of real-valued random functions. For a complex-valued function . / 2 Lp .‚/ means that Ej . /jp < 1,

2 ‚, i . e.,

 2 Lp .‚/

and

2 Lp .‚/.

The correlation function for a complex-valued random function is defined by    R. 1 , 2 / D E . 1 /  m. 1 / . 2 /  m. 2 / , where  stands for the complex conjugate of . The following properties of the correlation functions can be easily checked: 1. R. , /  0, where the equality appears if and only if the random function is constant with probability one. 2. R. 1 , 2 / D R. 2 , 1 /. 3. jR. 1 , 2 /j2  R. 1 , 1 /R. 2 , 2 /. 4. For any integer n, 1 , : : : n and complex numbers 1 , : : : , n , n X

R. j , k / j N k  0.

j ,kD1

Note that properties 1–3 follow from property 4. For two random functions 1 . / and 2 . / (belonging to Lp .‚/) one defines the cross-correlation function    R1 2 . 1 , 2 / D E 1 . 1 /  E 1 . 1 / 2 . 2 /  E. 2 / Extension to multidimensional complex-values random functions. Let 1 . /, 2 . /, : : : , r . / be a set of random complex-valued functions. It is considered as a complexvalued random function . / D . 1 . /, 2 . /, : : : , r . //T , 2 ‚. Here ./T stands for the transpose, so . / D . 1 . /, 2 . /, : : : , r . //T , 2 ‚ is a column. For two columns,  D .1 , 2 , : : : , r /T and D . 1 , 2 , : : : , r /T , we define a matrix   by 0 1 1 N 1 1 N 2 : : : 1 N m B2 N 1 2 N 2 : : : 2 N m C C.   D B @ ::: ::: ::: ::: A r N 1 r N 2 : : : r N m

13

Section 1.3 Fundamental concepts

For a random vector function . / D . 1 . /, : : : 1 . r //T we put m. / D .m1 . /, : : : , mr . //T D E . / D .E 1 . /, : : : , E r . //T,     D E Π. 1 /  m. 1 / Π. 2 /  m. 2 / R. 1 , 2 / D Rij . 1 , 2 / i,j D1,:::r ij   D EΠi . 1 /  mi . 2 / Πj . 2 /  mj . 2 / . ij D1,:::,r

The vector m. / is an r-dimensional complex-valued vector function; it is the expectation, called also a mean of the random function . /. The matrix R. 1 , 2 / is called a correlation matrix.

1.3.2

Gaussian random vectors

There is an important class of random functions completely defined by their first two moments, the expectation and correlation function. They are called Gaussian random functions and by definition have a Gaussian form of the finite-dimensional distributions. So we recall here first the case of random vectors. The Gaussian random vector  D .1 , 2 , : : : , n /T is defined by its characteristic ² ³ function 1 '.u/ D E ei.u,/ D exp i .m, u/  .Ru, u/ , (1.17) 2 where m D .m1 , : : : , mn /, u D .u1 , : : : , un /, R is a nonnegative definite real, n. Here we use the convenvalued symmetric matrix R D .rik /, i , k D 1, 2, : : : P n tional definition of a scalar product so that .m, u/ D kD1 mk uk and .Ru, u/ D Pn r u u . j ,kD1 j k j k The following statement explains the role of the above definition. The function

³ ² 1 .u/ D exp i.m, u/  .Ru, u/ 2

is a characteristic function of a random vector  if and only if the real-valued matrix R is non-negative definite and symmetric. The rank of the matrix R equals the dimension of a subspace where the distribution of the vector  is concentrated. If r, the rank of the matrix R is less than n, then the random vector is concentrated in an r-dimensional hyperplane; hence it has no density. Such a distribution is called a singular Gaussian distribution. If r D n, the random vector  has the density ² ³ 1 1 exp  .R1 .x  m/, .x  m// , (1.18) f .x/ D p 2 .2/n where R1 is the inverse of the matrix R,  D det.R/ is the determinant of R.

14

Chapter 1 Introduction

Let us give a series of statements which are well known from the probability theory, e. g., see [71]. 1. In the expression (1.17), m D .m1 , : : : , mn /T is the vector of expectations, and R is the correlation function: m D E, rj k D EŒ.j  mj /.k  mk /. 2. If the correlation function of a Gaussian vector  is not singular, then there exists an n-dimensional probability density f .x/ which is defined by (1.18). 3. The mutual distribution of any group of components of a Gaussian vector is Gaussian. 4. If  D .1 , : : : , n /T is a Gaussian vector, and random vectors  0 D .1 , : : : , r /T ,  00 D .rC1 , : : : , n /T are noncorrelated, then  0 and  00 are independent. 5. Gaussian distributions remain Gaussian under linear transformation. Let us give some other properties of the Gaussian distributions which are useful in practice. Assume we are given two vectors with Gaussian distributions:  D .1 , : : : , n /T and D . 1 , : : : , m /T . We are interested in the conditional distribution of the vector , assuming that is fixed. Without loss of generality we suppose that the correlation matrix R22 of the vector is nonsingular. Indeed, if R22 is singular, it means some components of are linearly dependent on the other components. Then, we exclude these components, and the dimension of is decreased. So let m1 D E, m2 D E , and let R11 be the correlation matrix of the vector , and R12 be the crosscorrelation matrix of the vectors  and : R12 D E.  m1 /  m2 / . Let us introduce 1 .  m /. Then the conditional distribution of the veca vector Q D m1 C R12 R22 2 tor , under the condition that is fixed, is Gaussian with the conditional expectation E.j / D Q and the conditional correlation matrix °   ˇ ± 1 R21 . E   Q   Q ˇ D R11  R12 R22 Notice the following important property: the matrix of the conditional correlations of the vector , fixed, is not random, and in particular, it does not depend on the value of .

1.3.3 Gaussian random functions A vector n-dimensional random function . / D ¹1 . /, : : : n . /º is called a Gaussian random function if the mutual distribution function of all components of the random vectors . 1 /, : : : , . n / is Gaussian. The correlation matrix R of the mutual distribution of the vectors . 1 /, : : : , . n / has a dimension sn sn and can be di-

15

Section 1.3 Fundamental concepts

vided in square blocks of size s s as follows: 0 R. 1 , 1 / R. 1 , 2 / B R. 2 , 1 / R. 2 , 2 / RDB @ ::: ::: R. n , 1 / R. n , 2 /

::: ::: ::: :::

1 R. 1 , n / R. 2 , n / C C, ::: A R. n , n /

where R. 1 , 2 / is the correlation matrix of the function . /. The reverse statement is true: for any real-valued vector function m. / and a nonnegative definite symmetric matrix function R. 1 , 2 / there exists an r-dimensional Gaussian random function for which m. / is the expectation, and R. 1 , 2 / is the correlation matrix. The Gaussian random functions play an extremely important role in many practical problems. This can be explained generally as follows. The real processes are usually affected by many random independent factors and the resulting superposition of these factors tends to a Gaussian distribution. This can be rigorously formulated as a limit theorem of normal correlations, which is a generalization of the well-known central limit theorem. Let us present this statement. A sequence of random functions n . /, 2 ‚, n D 1, : : : is said to be weakly convergent to a random function . /, 2 ‚ if for any s the mutual distribution of the series of random variables ¹ n . 1 /, : : : , . s /º is weakly convergent, as n ! 1, to the distribution of ¹. 1 /, : : : , . s /º. Theorem 1.1 (see [71]). Assume we are given a family of sums of random functions mn X

n . / D

˛nk . /,

2 ‚,

n D 1, 2 : : :

kD1

and the following conditions are satisfied: 1. For fixed n, the random variables ˛n1 . 1 /, ˛n2 . 2 /, : : : , ˛nm . m / are all mutually independent for each 1 , 2 , : : : m , and have finite second moments such that E˛nk . / D 0,

2 2 E˛nk . / D bnk . /.

2. The correlation function Rn . 1 , 2 / D EΠn . 1 / n . 2 /, converges to a limit: lim Rn . 1 , 2 / D R. 1 , 2 /.

n!1

3. The sums n . / D each  > 0

P mn

kD1

˛nk . / satisfy, for each , the Lindeberg condition: for

mn Z 1 X x 2 d…nk . , x/ ! 0, Bn2 jxj> Bn kD1

16

Chapter 1 Introduction

where …nk . , x/ is the distribution function of the random variable ˛nk . /, and Bn2 D

mn X

2 bnk . / D Rn . , /.

kD1

Then the random function n . / is weakly convergent, as n ! 1, to a Gaussian random function with zero expectation and correlation function R. 1 , 2 /.

1.3.4 Random fields Assume that our parameter is a point x 2 IRn . Let .x/ D . 1 .x/, : : : , d .x//T be a vector random function possibly with complex values, defined for all x 2 IRn . It is called a sl random field. Note that if n D 1, one uses the notion of random process. The random field .x/ is called homogeneous in a broad sense if E .x/ D m D const,

E. .x/  m/. .x/  m/ D R.x  y/,

where R.x/ is a continuous matrix function, the correlation of the homogeneous random field. The matrix function R.x/ is nonnegative definite. This means that for any d -dimensional complex vectors zk , points xk 2 IRm , k D 1, : : : , n, and for any integer n n X zk R.xk  xj /zj  0. k,j D1

A random field .x/, x 2 IRm is called meansquare continuous (m.s.c.) if from x.n/ ! x it follows that Ej.x.n/ /  .x/j2 ! 0 as n ! 1. The correlation functions and nonnegative definite functions are related by the following theorem. Theorem 1.2 (Bochner–Khinchin theorem). A matrix function R.x/, x 2 IRd is a correlation function of a homogeneous, m.s.c. random field if and only if it can be represented in the form Z ei.x,bf u/ F .d u/, (1.19) R.x/ D IRd

where F .A/ is a matrix-valued complex countable-additive function defined on Borel sets in IRm such that z F .A/z  0 for any complex vector z and any Borel set A

IRd , and the trace Sp F .IRm / < 1. A random field .x/ is called isotropic if it is homogeneous and it correlation function R.x/ depends on the length jxj. Thus for an isotropic random field R.x/ D R./

17

Section 1.3 Fundamental concepts

1.3.5

Stochastic measures and integrals

In this section we introduce integrals with respect to stochastic measures, known as stochastic integrals. R Let . , F , P / be a probability space, Ef D dP is the expectation, L2 . / D L2 . , F , P / is a class of random variables with finite second moment. Let X be a set, and K be a semi-ring of the subsets of X . Assume that each  2 K is related to a complex-valued random variable ./ which satisfies the following conditions: 1. ./ 2 L2 . /, .;/ D 0, S T 2. .1 /2 / D .1 / C .2 /.mod.P /, if 1 2 D ;, T 3. E .1 / .2 / D m.1 2 /, where m./ is a function defined on the sets of K. A family of random variables ¹ ./,  2 Kº satisfying the conditions 1–3 is called an elementary orthogonal stochastic measure, and m./ is its structure measure. The T orthogonality property of the stochastic measure is expressed by condition 3: if 1 2 / D ;, then E .1 / .2 / D 0. From the definition of m./ follows that it is nonnegative: m./ D Ej ./j2  0, m.;/ D 0 T and additive, i. e., if 1 2 / D 0, then  [  m 1 2 D Ej .1 / C .2 /j2  \  D m.1 / C m.2 / C 2m 1 2 D m.1 / C m.2 /. Now, a class L0 ¹Kk of simple functions is introduced: f .x/ D

n X

cr  r .x/,

r 2 K,

r D 1, 2, : : : n,

(1.20)

rD1

where n is arbitrary, and A .x/ is the indicator of the set A, c1 , : : : cr are complex numbers. The stochastic integral of a simple function f .x/ 2 L0 .K/ with respect to a stochastic measure ./ is defined by the formula Z n X cr .r /. (1.21)

D f .x/ .dx/ D rD1

For any two functions f .x/, g.x/ 2 L0 .K/ the following equality holds: Z Z Z E f .x/ .dx/ g.x/ .dx/ D f .x/g.x/ N m.dx/.

(1.22)

18

Chapter 1 Introduction

Assume that m satisfies the semi-additive condition and hence can be prolonged to a complete measure ¹X , B, mº. Then L0 ¹Kº is a linear subset of the Hilbert space L2 .m/ D L2 ¹¹X , B, mº. Denote by L2 ¹Kº the close of L0 .K/ in L2 .m/. Now we introduce a linear span L0 ¹ º of the family of random variables ./,  2 Kº i. e., a set of random variables which can be represented in the form (1.21); the space L2 . / is defined as a closure of L0 . / in the Hilbert space of random variables L2 . , F , P /. Notice that the relation (1.21) establishes an isometrical mapping D .f / between L2 ¹Kº and L0 .K/ and L0 . /. This mapping can be prolonged to an isometry between R L2 ¹ º. If D .f /, f 2 L2 ¹Kº, then we define D .f / D f .x/ .dx/. The random variable is then called a stochastic integral of f with respect to the measure . From this follows the following. The following statement is true (see [71]. Theorem 1.3. (a) For a simple function .1.20/ the stochastic integral is defined by the formula .1.21/. .1.22/ holds. (b)ZFor any f and g from L2 ¹mº D LZ2 ¹X , B, mº, the equality Z   ˛ f .x/ C ˇ g.x/ .dx/ D ˛ f .x/ .dx/ C ˇ g.x/ .dx/. (c) (d) For an arbitrary sequence of functions f .n/ .x/ 2 L2 ¹X , B, mº such that Z jf .x/  f .n/ .x/j2 m.dx/ ! 0, the following relation is true: Z

Z f .x/ .dx/ D l.i.m.

n!1

f .n/ .x/ .dx/.

Here l.i.m. means a limit in mean square sense, i. e., l.i.m. n D  implies lim Ejn  n!1

j2 D 0.

n!1

The existence of a sequence of simple functions approximating an arbitrary function f .x0 2 L2 ¹X , B, mº follows from the general theorems of the measure theory. Thus the stochastic integral can be considered as a mean square limit of the relevant integral sums. Let us denote by B0 the class of all subsets A 2 B with m.A/ < 1, and define a Q random function of sets .A/ by Q .A/ D

Z

Z A .x/ .dx/ D

.dx/. A

(1.23)

19

Section 1.3 Fundamental concepts

We list the following properties of this function: Q 1. .A/ is defined on the class of sets B0 ; S T 2. if An 2 B0 , n D 0, 1, : : : , A0 D 1 Ar D ; for k ¤ r, k > 0, nD1 An , q Ak P1 Q Q then .A0 / D nD1 .An / in mean square sense; T Q Q 3. E .A/ .B/ D m.A B/, A, B 2 B0 ; Q 4. ./ D ./

for  2 K.

An orthogonal stochastic measure is defined as a random set function Q satisfying the above conditions 1–4. Note that property 4 means that Q is a prolongation of the elementary stochastic measure ./. Thus the following statement is true [71]. Theorem 1.4. If the structure measure of the elementary stochastic measure ./. is Q semi-additive, then ./ can be prolonged to a stochastic measure ./. Note that Z Z Q f .x/ .dx/ D f .x/ .dx/ Q since L2 ¹ º D L2 ¹ º.

1.3.6

Integral representation of random functions

Using the results of the previous section we can represent random functions via stochastic integrals. Let ‚ be an arbitrary parameter set, and . , F , P / a probability space. Assume first that a p-dimensional random vector function . /, 2 ‚/ can be written in the form Z . / D g. , x/ .dx/, (1.24) where ./ is a stochastic measure on a measurable space .X , B/ with its values in C p and a structure measure m./, and g. , x/ is a scalar function such that for any 2 ‚ g. , x/ 2 L2 .m0 / D L2 .X , B, m0 /,

m0 ./ D Sp m./ D

p X

mkk ./.

kD1

The correlation function reads B. 1 , 2 / D E. 1 /  . 2 / D

Z g. 1 , x/g. 2 , x/ m.dx/.

(1.25)

We recall that .X , B, m0 / is a space with a complete measure, L2 .m0 / D L2 .X , B, m0 / is a Hilbert space of b-measurable complex-valued functions which

20

Chapter 1 Introduction

are square m0 -integrable: ² ³ Z L2 .m0 / D f : X ! C : jf .x/j2 m0 .dx/ < 1 . By L2 ¹gº we denote the closure in L2 .m0 / of a linear span generated by the family of functions ¹g. , x/, 2 ‚º. Then, L2 ¹gº is a linear closed subspace of L2 .m0 /. If L2 ¹gº D L2 .m0 /, then the family of functions ¹g. , x/, 2 ‚º is called complete in L2 .m0 / D L2 .X , B, m0 /. Let ¹g. /, 2 ‚º be a Hilbert random function with its values in C p , and L0 ¹º is a set of all random vectors

D

n X

ck . k /,

n D 1, 2, : : : ,

k 2 ‚,

kD1

where ck are arbitrary complex numbers, and L2 ¹º is a closure of L0 ¹º in the sense of mean square convergence of random vectors. A family of random vectors ¹ ˛ , ˛ 2 Aº, ˛ 2 L2 . / is called subordinate to a random function . /, 2 ‚º if ˛ 2 L2 ./, ˛ 2 A. For random functions whose correlation function can be represented in the form (1.25) the following theorem can be formulated, see [71]. Theorem 1.5. Assume that the correlation matrix of a random function ¹. /, 2 ‚º can be written in the form .1.25/ where m is a positive definite matrix measure on .X , B/, g. , x/ 2 L2 .m0 /, 2 ‚ .m0 D Spm/, and the family ¹g. , x/, 2 ‚º is complete in L2 .X , B, m0 /. Then . / can be represented in the form .1.24/ where ¹ .B/, B 2 Bº is a stochastic orthogonal measure which is subordinate to the random function ¹. /, 2 ‚º with a structure measure m./, and the equality .1.24/ holds with probability one for each . Let .x/ D . 1 .x/, : : : ,  d .x//, x 2 IRm is centered mean square continuous homogeneous (in a broad sense) random field with real- or complex values. Let R.x/ be the correlation function of this field. By the Bochner–Khinchin theorem, this correlation function obeys the representation (1.19). This representation is a particular case of (1.25) in which X D ‚ D IRm , g. , x/ D exp ¹i.x, /º. Since the family ¹exp ¹i.x, /º, 2 IRm º is complete in L2 .m0 / D L2 .X , b, m0 / where m0 is an arbitrary bounded measure on the sigma-algebra B m of IRm , we come to the following corollary of the previous theorem which is known as the spectral theorem (see [71]). Theorem 1.6. Any centered mean square continuous homogeneous vector random field . /, 2 IRm can be represented in the form Z . / D ei.x,/ .d x/, 2 IRm ,

21

Section 1.3 Fundamental concepts

where .A/, A 2 B m is a vector orthogonal measure on B m which is subordinate to .0/. Between L2 ¹º and L2 ¹F º, F0 D SpF there is an isometry providing (a) . / 2 L2 ¹º $ e i.x,/ 2 L2 ¹F0 º; (b) if i $ gi .x/, i 2 L2 ¹º, gi 2 L2 ¹F0 º, i D 1, 2, then Z Z 

i D gi .x/ .d x/, E 1 2 D g1 .x/g2 .x/ F .d x/.

1.3.7

Random trajectories

A random trajectory can be defined as follows. A random trajectory is defined as a solution to an ordinary differential equation with a random field in the right hand side: dXi .t , !/ D fi .X.t , !/, t , !/, dt

t  t0 , ! 2 , i D 1, : : : , n,

(1.26)

where f .x, t / D f .x, t , !/ D .f1 .x, t , !/, : : : , fn .x, t , !//, x 2 IRn is a vector random field, X.t , !/ D .X1 .t , !/, : : : , X1 .t , !//. The samples of the random field f .x, t / are assumed to be smooth enough in the sense that for each sample the classical solution of the deterministic equation (1.26) exists. An important difference to the classical deterministic ordinary differential equation is only in that in the considered case the right-hand side depends on a parameter ! 2 where is a probabilistic space. So in contrast to the stochastic differential equations of the Ito type, the solution is well defined in the classical sense, and there is no need to develop a special calculus for studying (1.26) . However it does not imply that all the question about the solutions to (1.26) can be answered by adapting the relevant results of the classical theory of differential equations. For example, the existence of the solution of an initial value problem for (1.26) is not equivalent to the existence of the solution for all (or almost all) samples (i. e., all values of !), one needs the existence of the solutions as random processes on one common interval (not depending on !). Let us give an illustrating example. Assume we solve the following equation: dX D  X 2, dt

X 2 IR1 , t 2 Œ0, 1/,

where  D .!/ is a standard Gaussian random variable. The solution of this equation with the prescribed initial value X.0/ D x0 can be written explicitly as ´ 0, if x0 D 0, 0  t < 1 1 X.t , !/ D  1 , if jx0 j ¤ 0, 0  t < ı. x0  .!/t Here

² ı D ı.x0 , !/ D

1 ,  x0

1,

if x0  > 0 if x0  < 0

22

Chapter 1 Introduction

is the explosion time instant, i. e., lim t!ı jX.t / D 1. This means, that for any x0 ¤ 0, the solution may explode arbitrarily quickly, with positive probability, i. e., for any " > 0, we get P .ı < "/ > 0. This implies that the problem has no sample solutions. Another issue is the existence of statistical characteristics of the solution to (1.26). For example in the framework of the conventional theory of differential equations it is not possible to Ranswer the question of whether or not the first moment EjX.t /j is finite, i. e., when  jX.t , !/ P .d!/ < 1. Nevertheless, many results in stochastic differential equations of the type (1.26) were obtained (e. g., see [9, 37, 228]) via an extension of the relevant results of the classical differential equation theory. For example, for the existence of a sample solution to (1.26) on a finite interval, say, I D Œ0, 1, it is necessary to prove the existence of measurable random processes 1 .t , !/, 2 .t , !/, .t , !/ 2 I !, such that P ¹i .t / dt < 1º D 1, i D 1, 2, and almost all samples of the right-hand side f .x, t / D f .x, t , !/ should satisfy the linear growth condition jf .x, t /j  1 .t / C 2 .t / jxj

for all x 2 IRn .

This result follows from the classical theory of differential equations (e. g., see [75]).

1.3.8 Stochastic differential, Ito integrals As already mentioned above, the motion of fluid elements and aerosol particles are often described by Langevin stochastic differential equations, known in the theory of stochastic processes as Ito stochastic differential equations. We have given the definition of stochastic integrals with respect to general stochastic measure, and so we could just refer to a particular case of Wiener measure to define the Ito integrals. Let us however begin with considerations which are closer to physics.

1.3.9 Brownian motion Let us again consider the ordinary differential equation (1.26), but now the randomness is specifically entering this equation as an additive noise: dX D b.t , X t / C  .t , X t /  “random noise”, dt

(1.27)

where b and  are some given deterministic functions. So intuitively, the solution X t is a random process with some distribution and correlations caused by the input noise W t D “random noise”. Again from physical intuition, we might assume that W t has the following properties: (i) W t1 and W t2 are independent if t1 ¤ t2 ; (ii) W t is a stationary process, i. e., the joint distributions of ¹W t1 Ct , : : : , W tk Ct º do not depend on t ; (iii) EW t D 0 for all t . However, it turns out there does not exist any “reasonable” stochastic process satisfying (i) and (ii): such a process cannot have continuous trajectories (e. g., see [70]). If

Section 1.3 Fundamental concepts

23

we assume EjW t2 j D 1, then the function .t , !/ ! W t .!/ cannot even be measurable, with respect to the  -algebra B F where B is the Borel  -algebra on Œ0, 1/ (e. g., see [158]). So the functions W t belong to other class we considered above, generalized stochastic functions, constructed as a probability measure in the space of tempered distributions on Œ0, 1/, and not as a probability measure on the much smaller space IR0,1 , like an ordinary stochastic process. We will avoid this construction and turn to a description of stochastic differential equations which uses random processes with independent increments. Let us start with the physical process of Brownian motion. Let us consider a particle which is moving on a line: the particle starts from the origin .0/ D 0, and during a small time increment t it makes a jump to the left with probability 1=2, and to the right with probability 1=2. So during time t the particle makes n D t =ıt jumps. We put ´ x with probability 1/2, xi D x with probability 1/2. Pn The total displacement of the particle reads as .t / D iD1 xi . Assuming that xi are mutually independent and equally distributed, the variance (dispersion) can be found as n X t D.t / D D xi D n D xi D n.x/2 D .x/2 . t iD1

Let us consider two arbitrary times s and t , where s < t . Then, the first and second summands in .t / D Œ.t /  .s/ C .s/ are independent, and in addition, the displacement during the time interval .s, t / depends obviously on t  s, since physically the character of the motion is not changing in the time, hence, .t /  .s/ and .t  s/ are equally distributed. Therefore, D.t / D DŒ.t /  .s/ C D.s/ D D.t  s/ C D.s/, which implies that D.t / is a linear function of t : D.t / D t  2 , where  2 is a constant called a diffusion coefficient. Thus we have t .x/2 D n Dxi , D.t / D t  2 D t and due to the Gaussian distribution of xi   n X .t / 1 P 0 for k 2 i if max jqmr .k/j > 0 (where Q D .qmr /). Taking the real part of (2.37) we get the mr

stratified randomization model of the original real-valued random field uE .x/: ²h n0 N i X 1 X 1 Q0 .kij / cos ij  Q00 .kij / sin ij  ij uE N ,n0 .x/ D p p n0 pi .kij / iD1 j D1 i ³ h C Q00 .kij / cos ij C Q0 .kij / sin ij ij . (2.38) Here ij D 2kij  x, ¹kij º, i D 1, : : : , N ; j D 1, : : : , n0 are the same as in (2.36), and  ij , ij , i D 1, : : : , N ; j D 1, : : : , n0 is a family of mutually independent and independent of the set ¹kij º n-dimensional real-valued standard Gaussian random variables.

2.5

Fourier wavelet models

In the numerical implementation of (2.25) we have to: (i) choose the scaling and wavelet functions, (ii) evaluate the coefficients (2.23), and (iii) find a reasonable choice of the cut-off parameters m0 and m1 in the approximations 1 X

Fm./ .2m0 x 0

C j / j '

j D1 1 X

m0 b0 Cb2 X xc

Fm./ .2m0 x C j / j , 0

(2.39)

j Db0 Cb2m0 xc 1 X

mDm0 j D1

Fm. / .2m x

C j /  mj '

m1 X

m b1 Cb2 X xc

Fm. / .2m x C j /  mj ,

mDm0 j Db1 Cb2m xc

(2.40)

40

Chapter 2 Stochastic simulation of vector Gaussian random fields

where bac stands for the integer part of a, and b0 D b0 .m0 /, b1 D b1 .m/ are such in./ . / tegers that supports of the functions Fm0 and Fm belong essentially to the intervals Œb0 , b0  and Œb1 , b1 , respectively. We will deal here with a scalar random process u.x/, x 2 IR. Extensions to vector random processes is straightforward.

2.5.1 Meyer wavelet functions The Meyer wavelet functions .x/ and (e. g., see [36]): Z 1 O d k, e i 2 kx .k/ .x/ D 1

where

.x/ are defined by their Fourier transforms Z .x/ D

1

e i 2 kx O .k/ d k,

1

8 1 ˆ ˆ < O cosŒ 2 .3jkj  1/, .k/ D ˆ ˆ : 0, 8 e i  k sinŒ 2 .3jkj  1/, ˆ ˆ < O .k/ D e i  k cosŒ 2 . 32 jkj  1/, ˆ ˆ : 0,

(2.41)

jkj  1=3, 1=3  jkj  2=3

(2.42)

otherwise, 1=3  jkj  2=3 2=3  jkj  4=3

(2.43)

otherwise.

Here .x/ is a smooth function satisfying the following conditions: .x/  0 for x  0, .x/  1 for x  1, and .x/ C .1  x/ D 1 for 0 < x < 1. As an example of such a function, we consider a function .x/ D p .x/ depending on a positive parameter p (see [47]): ² ³ p1 X 4p1 p p p j p .x/ D .1/ Œx  xj C , Œx  x0 C C Œx  xp C C 2 p j D1

where xj D .1=2/Œcos...p  j /=p// C 1, and ŒaC D max.a, 0/. The function p is p  1 times continuously differentiable, therefore, choosing p sufficiently large, we can make the functions O and O smooth enough.

2.5.2 Evaluation of the coefficients Fm./ and Fm.

/

Here we give some technical details on the calculation of the functions (2.23) which in our case read Z 4=3 e 2i k  g.k/ d k, (2.44) fm ./ D 4=3

41

Section 2.5 Fourier wavelet models

./ O where g.k/ D 2m=2 F 1=2 .2m k/.k/ and g.k/ D 2m=2 F 1=2 .2m k/ O .k/ for Fm and . / Fm , respectively. We calculate this function on the grid of points j D  N2  C .j  1/, j D 1, : : : , N , where N is an even number and   0 is the grid step. In order to evaluate the truncated sums appearing in the Fourier wavelet representation (2.22), we must choose N=2  b. We approximate the integral (2.44) by a Riemann sum: Z a N X e 2i k  g.k/ d k ' k e 2i kl j g.kl /, (2.45) fm .j / D a

lD1

where

2a . N We use the same number of points N D 2r (where r is some positive integer) to discretize the integral as we have done in representing fm ./ in physical space, so that we can apply the discrete fast Fourier transform. We also clearly need the cut-off in the integral in (2.45) to satisfy a > 4=3 (with g.k/ set to zero whenever evaluated for jkj > 4=3). Finally, the use of the fast Fourier transform requires that the steps in physical and wavenumber space be related through k D 1=N . Indeed, simple transformations then yield

 N j kl D   C .j  1/ Œa C .l  1=2/k 2 N 1 j 1 1  l 1 .j  1/.l  1/ D  1  C , (2.46) 4 2 N 2 N hence ² ² ³ ³ N  1 X .j  1/.l  1/ Gl exp  2i , (2.47) fm .j / ' exp i .j  1/ 1  N N kl D a C .l  1=2/k,

l D 1, : : : , N ;

k D

lD1

²

³ N 1 l 1  , Gl D k g.kl / exp  2i 4 2 which is in the form of a discrete Fourier transform. The constraints imposed on the discretization of the integral (2.45) to obtain an expression amenable to fast Fourier transform imply the following sequence of choosing parameters. First a bandwidth value b is chosen according to the desired accuracy in the Fourier wavelet representation (2.22). Then a spatial resolution  for fm ./ is selected, either according to the grid spacing h on a prespecified set of evaluation points or such that fm ./ can be calculated accurately enough by interpolation from the computed values. (In any event, we must have  < 3=8.) Next, a binary power N D 2r 1 is chosen large enough so that 2b=N  . Then we set a D 2  , and discretize the integral (2.45) with step size k D 2a=N D 1=.N/. where

42

Chapter 2 Stochastic simulation of vector Gaussian random fields

2.5.3 Cut-off parameters In practice, we use the approximation to u.x/:

u

.F W /

.x/ D

m0 b0 Cb2 X xc

Fm./ .2m0 x C j / j 0

j Db0 Cb2m0 xc

C

m1 X

m b1 Cb2 X xc

Fm. / .2m x C j / mj .

(2.48)

mDm0 j Db1 Cb2m xc

Recall that the parameters b0 D b0 .m0 / and b1 D b1 .m/ are chosen from the ./ . / criterion that the supports of the functions Fm0 and Fm belong essentially to the intervals Œb0 , b0  and Œb1 , b1 , respectively. Let us first suggest general arguments to the choice of parameters m0 and m1 . Theoretically, m0 can be arbitrarily, and for a fixed m1 , it would be reasonable to take the value of m0 large enough, since the cost per sample is proportional to m1  m0 C 1. However practical calculations show that an important criterion for the choice of m0 is that the value 2m0 is comparable with k0 , a characteristic wave number scale. This characteristic scale could be defined for instance as a value k0 , for which the integrals R k0 R1 0 F .k/ d k and k0 F .k/ d k are compared in the order of magnitude. It turns out that for values of m0 for which 2m0 is essentially larger than k0 , the dependence of b0 D b0 .m0 / on m0 is exponential (see below), hence the cost of simulation is then increasing. When choosing the parameter m1 one should remember that in the approximation of u.x/ by (2.48) the contributions of random harmonics with wave numbers from k 2 IR : jkj  2m1 are not taken into account. Therefore, to accurately simulate the random field in the interval .lmin , lmax /, the value of m1 should be taken so that 2m1 > 1= lmin . Thus, suppose these arguments have helped us to choose some values of m0 and m1 . The next question then which arises is naturally the quantitative criterion of the quality of approximation (2.48). It is clear that for Gaussian processes, the main criterion is to have good accuracy in evaluation of the correlation function. Therefore, the correlation function of u.F W / should well approximate the true correlation function B.r/ D hu.x C r/u.x/i in an interval .lmin , lmax / depending on the problem to be solved. In what follows we assume that this interval is given. Thus with the fixed values of parameters m0 and m1 , the error of our approximation for the random field (2.48) is defined by ".m0 , m1 / D

sup

lmin  rlmax

jB.r/  B .F W / .r/j ,

(2.49)

where B .F W / .r/ D hu.F W / .x C r/u.F W / .x/i. The random numbers j and mj in

43

Section 2.5 Fourier wavelet models

(2.48) are mutually independent, and hence X B .F W / .r/ D Fm./ .2m0 r C j / Fm./ .j / 0 0 j : jj j_jj b2m0 xcjb0

C

m1 X

X

Fm. / .2m r C j / Fm. / .j /;

(2.50)

mDm0 j : jj j_jj b2m xcjb0

here a _ b D max¹a, bº. The cost to calculate the value of the random field u.F W / in one point is proportional to T .m0 , m1 / D b0 .m0 / C

m1 X

b1 .m/.

(2.51)

mDm0

Proposition 2.1. Assume that a spectral function F .k/ satisfies the condition (B.4)

˘ for some s > 0, and the function Q belongs to the Nikolskii–Besov space B11 .IR/,  > 1=2. Then the following estimation is true: sup sup jB.r/  hu.F W / .x C r/u.F W / .x/ij

x2IR r2IR



m1 0 00 X Cm Cm Cs 0 C , C 2 1 2 1 4m1 s b0 .m0 / mDm0 b1 .m/

(2.52)

0 00 depending on F , , O and O . with the relevant constants Cs , Cm and Cm 0

The proof is given in Appendix B of this chapter. This estimation provides a justification for the construction of Fourier wavelet approximations. Indeed, (2.52) shows that the right-hand side of this estimation can be made arbitrarily small by a successive choice of the parameters. For example, we could first take some value of m0 , then choose b0 .m0 / so that the second term in the righthand side is small enough. Then choose m1 so that the first term is small, and finally, choose b1 .m/, m D m0 , : : : , m1 so that the last sum in (2.52) is small. Unfortunately this approach cannot be practically used since it is difficult to cal0 00 and Cml . In addition, the estimation (2.52) may be culate the coefficients Cs , Cm 0l crude, and so the described choice of parameters may be not the best one. Therefore in practice, we recommend using a direct estimation of sup jB.r/  B .F W / .r/j starting r2IR from (2.50).

2.5.4

Choice of parameters

In this section we discuss some aspects of the practical choice of the parameters of our model. First we study the influence of the parameters m0 and b0 on the accuracy of ./ approximation. So let us consider the behavior of Fm .y/ under the change of m. By

44

Chapter 2 Stochastic simulation of vector Gaussian random fields c

c homogeneous z0 = 5 cm

800

homogeneous, z0=25 cm

1200

700

1000

600 800

x = 20 m

500

d

600

xd = 50 m

400

x =20 m d

300

xd=100 m

400

xd = 100 m

200

200

x =50 m d

100 −2

−2

−1

10

10

X/h

10

−1

10

X/h

Figure 2.1. Fm./ .y/ as a function of y, for different values of m. Left panel: F .k/ D right panel: F .k/ D e

.k/2

2 ; 1C.2k/2

.

O it follows that for large the definitions (2.23) and (2.42) of the functions F ./ and , negative values of m Z 2=3 ./ O Fm .y/ D e i 2 ky 2m=2 Q.2m k/.k/ dk 2=3

Z 

2=3

2=3

O e i 2 ky 2m=2 Q.0/.k/ d k D 2m=2 Q.0/.y/.

(2.53)

provided that Q.k/ is continuous at the point zero. It implies that for such values of m, the function F ./ .y/ behaves analogously to the scaling function .y/ (which decreases as jyjp as jyj ! 1, where p is a parameter characterizing the smoothness p of the function O such that O 2 B11 .IR/; see Appendix B, Lemma 2.2. For simplicity, and for the sake of a clear presentation, let us assume that F .k/ D 0 if jkj  k , for same k > 0. Then for positive numbers m such that 2m =3  k Z 1=3 ./ Fm .y/ D e i 2 ky 2m=2 Q.2m k/ d k 1=3 Z O e i 2 ky 2m=2 Q.2m k/.k/ dk C Z D

1=3jkj2=3 1

1

e i 2 k2

m y

2m=2 Q.k/ d k D 2m=2 G.2m y/. ./

(2.54)

Hence with the increase of m, the effective support of the function Fm is expanded exponentially. Experience tells us that such a broadening may happen in a more general case when the support of the spectral function is not compact. To illutrate this, we

45

Section 2.5 Fourier wavelet models ./

show in Figure 2.1 the plots of the functions Fm .y/ for different values of m, for 2 F .k/ D 2=.1 C .2k/2 / (see the left panel) and F .k/ D e .k/ (right panel). So this suggests that the parameter m0 should be chosen so that at m D m0 , the ./ exponential broadening of the support of Fm has not yet happened. This implies the m condition 2 0  k0 , where k0 is a characteristic wave number scale. For these values ./ of m0 the width of the support of Fm0 is practically independent of m0 (recall that for m ! 1 the support width is defined by the function , in view of (2.53)). As a reasonable approximation we can put b0 D 10. . / Let us analyse the behavior of Fm .y/ for different values of m. The main differ./ . / ence between this function and the function Fm is that Fm is determined by the ./ values of Q.k/ for 2m =3  jkj  2.mC2/ =3; while Fm depends on the values of Q.k/, jkj  2.mC1/ =3. Let us see what the influence of this difference is. Take an example of a spectrum F .k/, which for jkj  k0 has the form F .k/ D CF jkj˛ ,

jkj  k0

(2.55)

for some k0 > 0 and ˛ > 1. Then, for 2m =3  k0 , we conclude by the definitions (2.23) and (2.43) of the functions F . / and O , respectively that for m  log2 .3k0 / Z Fm. / .y/ D e i 2 ky 2m=2 Q.2m k/ O .k/ d k 1=3jkj4=3 m

D 2 2 .˛1/ CF A˛ .y/, 1=2

(2.56)

where Z

˛ e i 2 ky jkj 2 O .k/ d k.

A˛ .y/ D 1=3jkj4=3

. /

It follows from this equality that the support of Fm does not broaden as m increases. Our experience shows that this property holds for many different spectral functions . / F .k/. We illustrate this in Figure 2.2, where Fm .y/ for F .k/ D 2=.1 C .2k/2 / 2 (the left panel) and F .k/ D e .k/ (right panel) are presented. It is seen that for . / both spectral functions, Fm .y/ are practically defined on 10  y  10. So we put b1 D 10 for any m. Thus when the values of b0 and b1 are fixed, we can study the dependence of the Fourier wavelet model on m0 and m1 . As an example, let us consider the case F .k/ D 2=.1C.2k/2 / with the relevant correlation function B.r/ D exp.r/. Since k0 ' 1, we can put m0 D 0 (recall that 2m0 should be of the order k0 ). In the first two rows of Table 2.1 the dependence of the error (2.49) on m1 is given, for the fixed value m0 D 0, and lmin and lmax are chosen as lmin D 0 and lmax D 5.

46

Chapter 2 Stochastic simulation of vector Gaussian random fields C

C

xd = 100 m

180

160

x = 20 m

160

d

140

x = 50 m d

x = 50 m d

140 120

xd = 20 m

120

x = 100 m

100

x = 200 m

homogeneous z = 5 cm 0

d

100

xd = 200 m

80

d

80 60

60

homogeneous, z0 = 25 cm

40

40 20

20

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

X/h

X/h

Figure 2.2. Fm. / .y/ as a function of y, for different values of m. Left panel: F .k/ D 2 right panel: F .k/ D e .k/ .

2 ; 1C.2k/2

Table 2.1. The errors of the Fourier-wavelet models, for different values of m0 and m1 . F .k/ D 1=.1 C .2k/2 /, lmin D 0, lmax D 5. m1

1

2

3

4

5

6

7

".0, m1 /

0.0455

0.0233

0.0121

0.0068

0.0041

0.0029

0.0024

m0

3

4

5

6

7

8

9

".m Q 0 , 3/

0.2714

0.1464

0.0789

0.0476

0.0303

0.0217

0.0124

It can be seen that the error rapidly decreases with m1 , and for m1 D 3 it is about 0.0121. In this example, we have compared our Fourier wavelet model (2.48) with the Fourier wavelet model [47] which can be obtained by ignoring the first sum in (2.48). Note that the choice of m0 in the model [47] is more difficult. In the last two rows of Table 2.1 we show the error ".m Q 0 , m1 / of the model [47], for different values of m0 , for fixed m1 D 3. The cost of calculation of u.F W / is proportional to m1  m0 (provided that b0 and b1 do not depend on m/, so the results of Table 2.1 show that in this example the cost of the method [47] is more than twice that of the model (2.48). In all the above calculations we have fixed the parameter values as b0 D b1 D 10. Let us now examine the dependence of the error of the model (2.48) on the parameters b0 and b1 . For the case of F .k/ D 1=.1 C .2k/2 / we have reached the accuracy ".0, 6/ D 0.01 with b0 D b1 D 5, and ".0, 6/ D 0.0029 was obtained with b0 D b1 D 6. This shows that the values b0 D b1 D 10 we recommended are in this case slightly overestimated.

Section 2.6 Fourier wavelet models of homogeneous random fields

47

We have also studied the error ".m0 , 6/ for the model (2.48) for different values of m0 . We have fixed b1 D 10, and the parameter b0 D b0 .m0 / was varying depending on m0 in such a way that ".m0 , 6/ ' 0.01. The computations give b0 .0/ D 5, b0 .1/ D 7, and then, b0 .m0 / is exponentially increasing as b0 .m0 C 1/ D 7  2m0 , m0  0. Hence, the cost of evaluation of the field (2.48) in one point is proportional to T .m0 , 6/ ' 7  2m0 1 C 10 .m1  m0 C 1/ (see (2.51)). If we choose b0 to ensure ".m0 , 6/ ' 0.003, then b0 .0/ D 6, b0 .1/ D 9, and b0 .m0 C 1/ D 9  2m0 , m0  0. This shows that the cost exponentially increases with m0 . Note that for vector processes uE .x/ D .u1 .x/, : : : , ul .t //T , x 2 IR (l > 1) the FW model is constructed analogously to the scalar case, as we obtained (2.48) from (2.25) by taking a finite number of terms. The following generalization of Proposition 2.1 can be given. Proposition 2.2. Assume that a spectral tensor F .k/ satisfies the condition Z

1

1

spF .k/.1 C jkj2 /s d k < 1 .spF D Fjj /

for some s > 0, and the entries of the tensor Q defined by (2.3) belong to the Nikol ˘ skii–Besov space B11 .IR/,  > 1=2. Then the following estimation is true: sup sup

.F W /

max jBij .r/  hui

x2IR r2IR i,j D1,:::,l

.F W /

.x C r/uj



.x/j

m1 0 00 X Cm Cm Cs 0 C , C 2 1 2 1 4m1 s b0 .m0 / mDm0 b1 .m/

0 00 depending on F , , O and O . and Cm with the relevant constants Cs , Cm 0

2.6

Fourier wavelet models of homogeneous random fields based on randomization of plane wave decomposition

2.6.1

Plane wave decomposition of homogeneous random fields

Our presentation of Fourier wavelet expansions was given for the 1-dimensional case, i. e., uE .x/, x 2 IR. Direct generalizations for the high-dimensional case are possible on the basis of relevant multidimensional scaling and wavelet functions; see, e. g., [231, 264]. However, with the dimension the complexity of this kind of models increases drastically. Therefore, it is reasonable to try to construct high-dimensional models via 1-dimensional ones.

48

Chapter 2 Stochastic simulation of vector Gaussian random fields

Here we present the plane waves decomposition method for simulating random fields using samples of relevant random processes. This approach is especially efficient for isotropic random fields; see, e. g., [47, 136]. Denote by Sd 1 a unit sphere in IRd , and let Ad be its surface area, d  the area element. Let us introduce a function B : IR Sd 1 ! IRnn such that B.; / is a correlation tensor for almost all  2 Sd 1 . We introduce also a n-dimensional Gaussian random field Z.t ; A/, t 2 IR, and A is a measurable subset of Sd 1 which is defined by hZ.t ; A/i D 0;

Z.t ; A1 [ A2 / D Z.t ; A1 / C Z.t ; A2 / for A1 \ A2 D ; (2.57) Z B.t1  t2 ; / d  (2.58) hZ.t1 ; A1 /ZT .t2 ; A2 /i D A1 \A2

for each t1 , t2 2 IR and measurable subsets A1 , A2 , A from Sd 1 . The block matrix B with the entries Z B.ti  tj ; /, i , j D 1, : : : , N Bij D Ai \Aj

(2.59)

is positive definite and symmetric for all positive integers N and values t1 , : : : tN in IR, and measurable subsets A1 , : : : , AN in Sd 1 . The proof of this statement will be presented in Appendix A of this chapter. Thus the Gaussian random field Z.t ; A/ is well defined. Note that Z.t ; / is a Gaussian white noise measure for fixed t . Let Hi ./, i D 1, 2 be R complex-valued l n-matrices, depending on  2 Sd 1 satisfying the condition kHi k2 d  < 1. We will use the following property: Z Sd 1

Z H1 ./ Z.t1 ; d /

 H2 ./ Z.t2 ; d / Sd 1 Z H1 ./B.t1  t2 ; /H2 ./ d . (2.60) D Sd 1

Given the correlation tensor in  , B. ; /, the spectral tensor in , Fv .; / is defined by Z 1 e i 2  B. ; / d  . (2.61) Fv .; / D 1

Let H./ be a l n matrix defined on Sd 1 . We define uEQ .x/ by uEQ .x/ D

Z Sd 1

H./Z.x  ; d /.

(2.62)

Section 2.6 Fourier wavelet models of homogeneous random fields

49

Here the stochastic integral is taken over the white noise measure Z.t ; A/ (e. g., see [70]). We show now that ˛ ˝Q uE .x C rE/uEQ T .x/ Z 1 O v .k, k/H O  .k/ O C H.k/F O v .k, k/H O  .k/º O d k, e i 2 rEk d 1 ¹H.k/F D d k IR (2.63) where k D jkj, kO D k=k. This implies that the spectral tensor FQ .k/ of the random field uEQ .x/ is given by 1 O v .k, k/H O  .k/ O C H.k/F O v .k, k/H O  .k/º. O ¹H.k/F k d 1

FQ .k/ D D

(2.64)

By the definition, we get from (2.60) E uEQ .x C rE/uEQ T .x/ DZ Z  E D H./ Z..x C rE/  ; d / H./ Z.x  ; d / S Sd 1 Z d 1 H./B.Er  ; /H  ./ d  D Z D

Sd 1

Z

d

Sd 1

Z D

Sd 1

d

Z D

Z

Sd 1

Sd 1

e i 2  rE H./Fv .; / H  ./ d 

1 Z 1

e i 2  rE H./Fv .; / H  ./ d 

0

Z

C

1

d Z

d

0 1

1

e i 2  rE H./Fv .; / H  ./ d 

e i 2  rE

0

® ¯ H./Fv .; / H  ./ C H./Fv .; / H  ./ d  Z 1 O v .k, k/H O  .k/ O C H.k/F O v .k, k/H O  .k/º O d k. e i 2 rEk d 1 ¹H.k/F D d k IR (2.65) Let uE .x/, x 2 IRd be a zero mean vector homogeneous Gaussian random field of dimension l defined by its spectral tensor F .k/. We are in position now to construct the random field uE .x/ as a superposition of the plane waves. Indeed, let Fv .; / D

jjd 1 F . /, 2

H./ D I,

(2.66)

50

Chapter 2 Stochastic simulation of vector Gaussian random fields

hence it follows from (2.63) that the random field Z Z.x  ; d / uE .x/ D

(2.67)

Sd 1

has the desired spectral tensor F .k/. Thus, we give an integral representation of homogeneous Gaussian random fields through a superposition of plane waves Z.x; d / (see (2.62) and (2.67)). A numerical model can be constructed through an approximation of the integral over an unit sphere by a finite sum. Here one can use both deterministic and stochastic approaches.

2.6.2 Decomposition with fixed nodes Let i , i D 1, : : : , Ns be a finite partition of the unit sphere Sd 1 : Sd 1 D s [N iD1 i , and i \ j D ; when i ¤ j . Choosing nodes i 2 i , i D 1, : : : , Ns , the following deterministic approximation to (2.62) can be constructed: uEQ .x/ D

Z Sd 1

H./Z.x  ; d / '

Ns X

H.i /Z.x  i ; i /,

(2.68)

iD1

where by definition (see (2.57) and (2.58)) Z.t ; i /, t 2 IR, i D 1, : : : , Ns are mutually independent zero mean stationary Gaussian processes with the following correlation structure: Z T B. ; / d  hZ.t C  ; i /.Z.t ; i // i D Z D

i

i

Z

1

1

e i 2  Fv .; /d d .

tensor given From R R this it follows that the process Z.t ; i /, t 2 IR has a spectral by i Fv .; /d  ' ji jFv .; i /, where ji j D i d . Hence

the process Z.t ; i / can be approximated by ji j1=2 vE.i/ .t / where vE.i/ .t / D .i/ .i/ .v1 .t /, : : : , vn .t //T is a random process with the spectral tensor Fv .; i /. From this we get by (2.68) the following approximation to the random field uEQ .x/, x 2 IRd : uEQ det Ns .x/ D

Ns X

ji j1=2 H.i /E v .i/ .x  i /.

(2.69)

iD1

By the construction it is clear that the smaller the subregions i , i D 1, : : : , Ns the better uEQ det Ns .x/ approximates the random field with the spectral tensor (2.64). In particular the model uE det Ns .x/ D

Ns X iD1

ji j1=2 vE.i/ .x  i /,

(2.70)

Section 2.6 Fourier wavelet models of homogeneous random fields .i/

51

.i/

where vE.i/ .t / D .v1 .t /, : : : , vl .t //T , i D 1, : : : , Ns are mutually independent zero d 1

mean Gaussian processes with spectral tensor F .i/ ./ D jj 2 F .i /, is an approximation of the random field uE .x/ D .u1 .x/, : : : , ul .x//T , x 2 IRd having the spectral tensor F .k/. Let us consider the case when the spectral tensor F .k/ has the form O O  .k/ F .k/ D H.k/H

2E.k/ , Ad k d 1

(2.71)

where H./,  2 Sd 1 is an l n-matrix satisfying the condition H./H  ./ D H./H  ./,

(2.72)

and E.k/ is a scalar nonnegative even function, Ad is the area of the unit sphere in IRd . Thus the approximation to the random field uE .x/ D .u1 .x/, : : : , ul .x//T , x 2 IRd with the spectrum (2.71) can be constructed by the formula uE det Ns .x/

D

Ns  X ji j 1=2 iD1

.i/

Ad

H.i /E v .i/ .x  i /

(2.73)

.i/

where vE.i/ .t / D .v1 .t /, : : : , vn .t //T , i D 1, : : : , Ns are mutually independent zero mean stationary Gaussian random processes with independent components each having the same spectral function Fv ./ D E./: Z 1 .i / .i / e i 2  E./ d . hvj11 .t C  /vj22 .t /i D ıi1 i2 ıj1 j2 1

s Let us give an example of subdivision on a 3D sphere, S2 D [N iD1 i . So, to 3 choose a subdivision of the unite sphere in IR , we work in the spherical coordinates . , /, 0   , 0    2. First we fix an integer parameter n  1 which defines the step  D =n and the relevant altitude mesh j D .j  1=2/ , j D 1, : : : , n . For each fixed j , .1  j  n /, we construct the latitude mesh jr D

.j /

.j /

.r  1=2/j , r D 1, : : : , n , where n n

.j /

D b2 sin. j /= c, j D 2=n .

.j /

 Finally, a 2-index subdivision ¹¹j ,r ºrD1 , j D 1, : : : , n º of the unit sphere S2 , ² ³   j j j ,r D . , / : j   < j C ; jr    < jr C , 2 2 2 2 (2.74)

P  .j / n surface elements. In is constructed, which consists of Ns D Ns .n / D jnD1 our calculations, we used the values of n and Ns as given in Table 2.2. We choose

52

Chapter 2 Stochastic simulation of vector Gaussian random fields

Table 2.2. Dependence of Ns on n . n

4

6

8

10

16

30

Ns

20

44

78

124

320

1132

the nodes as follows: j ,r D .sin. j / cos.jr /, sin. j / sin.jr /, cos. j //, .j /

j D 1, : : : , n , r D 1, : : : , n . (2.75) Thus the 3D FWM based on the plane wave decomposition with deterministic nodes is

n X   X jj ,r j 1=2 n

uE det n .x/ D where

.j /

j D1 rD1

4

H.j ,r / vE.jr/ .x  j ,r /,

(2.76)

Z jj ,r j D

j ,r

d  D 4j Œcos. j   =2/  cos. j C  =2/.

2.6.3 Decomposition with randomly distributed nodes A randomized model of the field uEQ .x/, x 2 IRd defined by the stochastic integral (2.62) with the spectral tensor (2.64) can be represented in the form 1=2 Ns Ad X .x/ D H.!i /E v .i/ .x  !i /, uEQ rnd Ns 1=2 Ns iD1

(2.77)

where !i , i D 1, : : : , Ns is a family of independent unit isotropic vectors in IRd , .i/ .i/ vE.i/ .t / D .v1 .t /, : : : , vn .t //T , i D 1, : : : , Ns is a set of mutually independent and stochastically independent of !i , i D 1, : : : , Ns , zero mean, stationary Gaussian random processes with the spectral tensor Fv ., !i /. In the general case of anisotropic random fields, the following generalization of the model (2.77) can be given: uEQ rnd Ns .x/ D

1

Ns X

1=2 Ns iD1

1 p 1=2 .!

i/

v .i/ .x  !i /, H.!i /E

(2.78)

where p./,  2 Sd 1 is a probability density function defined on a unit sphere Sd 1 , and the random points !i , i D 1, : : : , Ns are independently sampled on Sd 1 according to the probability density p./; the family vE.i/ .t / is constructed the same as in (2.77).

Section 2.6 Fourier wavelet models of homogeneous random fields

Proposition 2.3. Suppose that the density p./ satisfies the condition Z 1 Fv .; / d  > 0. p./ > 0 if 1

53

(2.79)

Then for any Ns the function uEQ rnd Ns .x/ defined by (2.78) is a zero mean homogeneous random field (generally non-Gaussian) with the spectral tensor (2.64): Z rnd rnd T Q Q huE Ns .x C rE/.uE Ns .x// i D e i2 rEk FQ .k/ d k. (2.80) IRd

Proof. The terms in the sum of (2.78) are independent; also, the process vE.i/ .t / is independent of !i , and hence T EQ rnd EQ rnd .x C rE/.uEQ rnd .x//T i huEQ rnd 1 1 Ns .x C rE/.u Ns .x// i D hu T D hH.!/hE v! ..x C rE/!/E v! .x!/j!iH  .!/i,

where ! is a random point on Sd 1 having the density p./. The random process vE! .t / is stochastically independent of !, having a spectral tensor Fv .; !/. Here hj!i means a conditional averaging under a fixed !. By this construction, Z 1 T v! .x!/j!i D e i2 rE! Fv .; !/ d . hE v! ..x C rE/!/E 1

Therefore, T hH.!/hE v! ..x C rE/!/E v! .x!/j!iH  .!/i Z Z 1 D d e i 2  rE H./Fv .; / H  ./ d . 1

Sd 1

Then we can proceed as we have done after the second row in (2.65); this leads us to the desired relation (2.80). Assume that the spectral tensor of the random field to be simulated has a representation (2.71) where the tensor H satisfies (2.72). Then, it is reasonable to choose p./  1=Ad , which means the points on !i are distributed uniformly; hence the simulation formula can be written as follows: uE rnd Ns .x/ D .i/

.i/

1

Ns X

1=2 Ns iD1

H.!i /E v .i/ .x  !i /,

(2.81)

where vE.i/ .t / D .v1 .t /, : : : , vm .t //T , i D 1, : : : , Ns are mutually independent zero mean stationary Gaussian random processes with independent components having the same spectral function E./.

54

Chapter 2 Stochastic simulation of vector Gaussian random fields

Decomposition with stratified randomly distributed nodes It is possible to construct a stratified randomization model by choosing a subdivision s ¹i ºN iD1 . Then in each i a random point !i 2 i is sampled. For simplicity, we may assume that it is uniformly distributed in i . So under the condition (2.71) we can use, along with the models (2.73) and (2.81) the following model with stratified randomly distributed nodes: uE srm Ns .x/

D

Ns  X ji j 1=2 iD1

Ad

H.!i /E v .i/ .x  !i /.

(2.82)

A hybrid stratified randomized FWM based on the plane wave decomposition (2.82) and subdivision sets (2.74) in 3D case has the form n X   X jj ,r j 1=2 n

uE srm n .x/ D

.j /

j D1 rD1

4

H.!j ,r / vE.jr/ .x  !jr /,

(2.83)

where !jr D .sin. Qj / cos.Qjr /, sin. Qj / sin.Qjr /, cos. Qi j //,   Qj D arccos .1  j / cos. j   =2/ C j cos. j C  =2/ , Qjr D jr C .jr  1=2/j , .j /

and j , jr , .j D 1, : : : , n ; r D 1, : : : , n / are mutually independent random numbers uniformly distributed on Œ0, 1.

2.6.4 Some examples Isotropic vector fields A homogeneous d -dimensional vector random field uE .x/, x 2 IRd is called isotropic if the random field vE.x/ D U T uE .U x/ has the same finite-dimensional distributions as those of the random field uE .x/ for any rotation U with transpose U T (e. g., see [146]). It is known that the general structure of the spectral tensor of an isotropic random field is the following [146]: ² ³  2 ki kj  ki kj C E .k/ ı  .k/ Fij .k/ D E , (2.84) 1 ij 2 k2 k2 Ad k d 1 where k D jkj, Ad is the area of the unit sphere in IRd , E1 and E2 are nonnegative scalar even functions.

Section 2.6 Fourier wavelet models of homogeneous random fields

55

The tensor (2.84) can be represented in the form F .k/ D

2E1 .k/ O  .k/ O C 2E2 .k/ H2 .k/H O  .k/, O H1 .k/H 1 2 d 1 Ad k Ad k d 1

.1/

.2/

(2.85)

.2/

where H1 D .hij /di,j D1 and H2 D .h1 , : : : , hd / are d d and 1 d matrix functions defined on the unite sphere Sd 1 by .1/

hij ./ D ıij  i j ; i , j D 1, : : : , d ,

.2/

hj ./ D j ,

 D . 1 , : : : , d / 2 Sd 1 .

The tensors Hi , i D 1, 2 obviously satisfy (2.72). Thus using the decomposition with fixed nodes (2.73), the isotropic random field uE .x/ can be approximated by uE det Ns .x/ D

Ns   X ji j 1=2  .i/ .i/ H1 .i /E v1 .x  i / C H2 .i /E v2 .x  i / , (2.86) Ad iD1

.i/

.i/

where vE1 , and vE2 i D 1, : : : , Ns are mutually independent zero mean, stationary d dimensional, Gaussian random processes with independent components. Each compo.i/ .i/ nent of vE1 .t / has the spectral function E1 ./, and each component of vE2 .t / has the .i/ .i/ v2 .t / D i  vE2 .t /. It is not difficult spectral function E2 ./. Let v .i/ .t / D H2 .i /E to check that the spectral function of the scalar process v .i/ .t /, t 2 IR is E2 ./. Thus we can present the formula (2.86) in a form convenient for simulation: Ns   X ji j 1=2  .i/ .i/ H .x/ D . /E v .x   / C v .x   / , (2.87) uE det 1 i 1 i i Ns Ad iD1

where v .i/ .t /, i D 1, : : : , Ns is a set of scalar random processes which are indepen.i/ dent of each other and of vE1 .t /, i D 1, : : : , Ns with the spectral function E2 ./. Analogously, using the decomposition with randomly distributed nodes (2.77), a d dimensional isotropic random field with a spectral density (2.84)) can be simulated by uE rnd Ns .x/ D

1 1=2

Ns

Ns   X .i/ H1 .!i /E v1 .x  !i / C v .i/ .x  !i / ,

(2.88)

iD1

where !i , i D 1, : : : , Ns is a family of independent unit isotropic vectors in IRd ; .i/ vE1 and v .i/ i D 1, : : : , Ns are mutually independent and independent of !i , i D 1, : : : , Ns , zero mean, stationary d-dimensional scalar Gaussian random processes with .i/ independent components. Components of vE1 .t / are independent of each other and have the same spectral function E1 ./ while the random process v .i/ .t / has the spectral function E2 ./. Note that the representation (2.88) is given in [136].

56

Chapter 2 Stochastic simulation of vector Gaussian random fields

Remark 2.1. Note that the simulation formulas (2.87) and (2.88) can be rewritten in the 3D case in different forms which involve vector products, as is done in [102, 191]. Indeed, the tensor .!/ with entries ij .!/ D ıij !i !j , ! 2 S2 can be represented as .!/ D H .!/H T .!/ where H .!/ is 3 3 tensor with entries hi i .!/ D 0, i D 1, 2, 3, and h12 .!/ D h21 .!/ D !3 , h13 .!/ D h31 .!/ D !2 , h23 .!/ D h32 .!/ D !1 . Taking into account H .!/ vE D ! vE, we rewrite the representations (2.87) and (2.88) in the form Ns   X ji j 1=2  .i/ i vE1 .x  i / C v .i/ .x  i / , .x/ D (2.89) uE det Ns 4 iD1

uE rnd Ns .x/ D

1 1=2

Ns

Ns   X .i/ !i vE1 .x  !i / C v .i/ .x  !i / .

(2.90)

iD1

2.6.5 Flow in a porous media in the first order approximation For saturated porous media flows, under time independent conditions, the so-called Darcy’s velocity, or specific discharge, q is determined by the Darcy law:   q.x/ D .x/E u.x/ D K.x/r .x/ , (2.91) where uE is the pore velocity, , the porosity, , the hydraulic potential  D z C pg=, , the fluid density, and K is the hydraulic conductivity. The functions and K are key parameters of the flow. Experimental measurements show a highly heterogeneous behavior of K in space with the following remarkable property: when considering K as a random field, its distribution is well approximated by the log-normal law (e. g., see [34,65]). Therefore, in models, the hydraulic log-conductivity Y D ln K is commonly considered to be a statistically homogeneous Gaussian random field . We denote Y 0 D Y  < Y >, CY .Er / D hY 0 .x C rE/Y 0 .x/i the auto-correlation function of the hydraulic log-conductivity. The porosity is also often considered in some models as a random field. However its variability is generally much smaller than that of K, and it is usually assumed to be constant. In the framework of the method of small perturbations, when the variation of the hydraulic log-conductivity is small, the spectral tensor F .k/ of the velocity field uE .x/ in a 3D porous medium within a first order approximation can be represented by the form (see, e. g., [65]) T O O O FY .k/, F .k/ D .k/U.. k/U/

(2.92)

57

Section 2.6 Fourier wavelet models of homogeneous random fields

where U D .U1 , U2 , U3 /T D hE ui is the mean velocity, Z O FY .k/ D e i 2 kEr CY .Er /d rE IR3

O is a 3 3 tensor is the spectral function of the hydraulic log-conductivity, and .k/ defined in the previous subsection. Assume that the hydraulic log-conductivity field Y D ln K is isotropic which implies that FY .k/ D FY .k/. Introducing E.k/ D 2k 2 FY .k/ and taking into account T O O F .k/ D .k/U.. k/U/

2E.k/ 4k 2

we get the following representation with fixed nodes for the velocity field uE .x/: uE det Ns .x/

DUC

Ns  X ji j 1=2 Ad

iD1

.i / U v .i/ .x  i /,

(2.93)

where v .i/ .t /, i D 1, ..., Ns are mutually independent zero mean, stationary Gaussian scalar processes with the spectral function E.k/. Analogously, using a decomposition with randomly distributed nodes (e. g., see (2.77)), the fluctuations of a 3-dimensional random field with the spectral density (2.92) can be simulated by uE rnd Ns .x/

DUC

1

Ns X

1=2

Ns

.!i / U v .i/ .x  !i /,

(2.94)

iD1

where !i , i D 1, : : : , Ns is a family of independent unit isotropic 3D vectors , v .i/ .t /, i D 1, : : : , Ns is a set of mutually independent and independent of !i , i D 1, : : : , Ns , zero mean, stationary Gaussian scalar processes with the spectral function E.k/ D 2k 2 FY .k/.

2.6.6

Fourier wavelet models of Gaussian random fields

We have presented above an approximation of a homogeneous Gaussian random field obtained by a superposition of plane waves. This approach makes it possible to simulate a random field uE .x/ through simulation of random processes vE.i/ .t /, i D 1, ..., Ns . If in the simulation of these random processes we use the Fourier wavelet method presented in Section 2.5, we obtain a Fourier wavelet model of Gaussian random fields based on the plane wave decomposition. If the random field is isotropic, or if its spectral tensor can be represented in the form (2.71) where the tensor H satisfies the condition (2.72), then the random processes vE.i/ .t /, i D 1, ..., Ns all have the same spectrum, which simplifies the construction of the simulation formulas.

58

Chapter 2 Stochastic simulation of vector Gaussian random fields

In the general case when the random field is anisotropic, the general simulation formula (2.70) is used. In this case the random processes vE.i/ .t /, i D 1, ..., Ns may generally have different spectral functions. This complicates the simulations when ./ . / precalculating the functions Fm and Fm which will also depend on the index i : ./ . / Fm,i and Fm,i , i D 1, ..., Ns . To this point we make the following remark. Assume that the random field uE .x/ is anisotropic, but can be transformed to an isotropic field by a linear transformation of x. More precisely, assume that there exists a nonsingular matrix A such that the random filed uE 0 .x0 / D uE .Ax0 / is isotropic. Then to simulate the random field uE we first simulate an isotropic random field uE 0 .x0 /, and then use the representation uE .x/ D uE 0 .A1 x/.

2.7 Comparison of Fourier wavelet and randomized spectral models As mentioned in the introduction to this chapter, our experiments in the 1-dimensional case (d D 1) have shown that the randomized models are more efficient for evaluating ensemble averages than the FW models, or compatible in efficiencies, if multipoint statistics should be calculated (see [104]). The situation is opposite for spatial averaging. In this section we compare the Fourier wavelet model (FWM) and the randomized spectral model (RSM) in the 3-dimensional case (d D 3). We simulate a 3dimensional zero mean Gaussian isotropic incompressible random field uE .x/ with a spectral tensor Fij .k/ D

2E.k/  ki kj  ı ,  ij 4 k 2 k2

i , j D 1, 2, 3;

E.k/ D 

8.2k/4 1 C .2k/2

3 . (2.95)

It is known that the correlation tensor of isotropic incompressible fields is defined by BLL .r/, the longitudinal, and BN N .r/, transversal correlation functions (e. g., see [146]). In our case these functions can be represented explicitly (e. g., see [146]): BLL .r/ D hu1 .x C r, 0, 0/u1 .x, 0, 0/i D e r , BN N .r/ D hu2 .x C r, 0, 0/u2 .x, 0, 0/i D e r .1  r=2/.

(2.96)

The comparison of RSM and FWM is made by calculation of the correlation functions (2.96).

2.7.1

Some technical details of RSM

As a basis, we take the model (2.38) for the simulation of isotropic random field uE .x/. The spectral space is divided in N D 3 subsets i D ¹k : ai  jkj < bi º, i D 1, 2, 3

59

Section 2.7 Comparison of Fourier wavelet and randomized spectral models

with a1 D 0, b1 D a2 D 0.34, b2 D a3 D 0.8, b3 D 1. In each subset i , n0 points ki,j , j D 1, : : : , n0 are sampled with the same probability density: 1 Ci pi .k/ D , 4k 2 .1 C .2k/2 /

Z k 2 i ;

Ci D 1=

bi ai

dk . .1 C .2k/2 /

Note that the spectral tensor (2.95) can be written in the form (2.3) with Q.k/ D O f 1=2 .k/H .k/:  T O f 1=2 .k/H .k/ O ; F .k/ D f 1=2 .k/.H k/

f .k/ D 16

.2k/2 , .1 C .2k/2 /3

and the 3 3-dimensional antisymmetric matrix H ./,  2 S2 is defined in the preO D kO  (a b is a vector product vious section. For a 3-dimensional vector  H .k/ of vectors a and b), hence in view of (2.38) we come to the simulation formula n0  N X 1 X f .kij / 1=2 uE N ,n0 .x/ D p n0 pi .kij / iD1 j D1 ® ¯ .!ij  ij / cos. ij / C .!ij ij / sin. ij / , (2.97) where kij D kij !ij , !ij , i D 1, : : : , N ; j D 1, : : : , n0 are mutually independent 3dimensional unit isotropic vectors, and kij i D 1, : : : , N ; j D 1, : : : , n0 are random points (independent of each other and of !ij ) distributed in Œai , bi  with the density pi .k/ D Ci =.1 C .2k/2 /. Here ij D 2kij !ij  x, and  ij , ij , i D 1, : : : , N ; j D 1, : : : , n0 are mutually independent and independent of the family kij standard 3-dimensional Gaussian random variables (with zero mean and unity covariance matrix). The points kij in (2.97) are sampled isotropically, which does not imply that the samples of uE N ,n0 .x/ are isotropic in space, especially when n0 are relatively small. To improve the isotropic property, it is reasonable to use a stratified sampling over angles using P the subdivision (2.74). Namely, we constructed the subdivision of the space IR3 D i,j ,r ijr , where ijr D ¹k 2 IR3 : ai  jkj < bi ; kO D k=jkj 2 j ,r º, .j /

where i D 1, : : : , N ; j D 1, : : : , n ; r D 1, : : : , n . In each element ijr we sample one point kijr D kijr !ijr , where kijr is chosen in Œai , bi / with the density pi .k/ D Ci =.1 C .2k/2 /, while the random direction !ijr is sampled uniformly in j ,r : !ijr D .sin. Qij / cos.Q ijr /, sin. Qij / sin.Q ijr /, cos. Qij //,   where Qij D arccos .1  ij / cos. j   =2/ C ij cos. j C  =2/ , Q ijr D jr C .j /

.ijr  1=2/j . Here ij , ijr (i D 1, : : : , N ; j D 1, : : : , n ; r D 1, : : : , n ) are

60

Chapter 2 Stochastic simulation of vector Gaussian random fields

mutually independent random numbers uniformly distributed in Œ0, 1. The random point kijr is distributed with the density 1 Ci . jjr j jkj2 .1 C .2jkj/2 /

pijr .k/ D

Thus the RSM for an isotropic Gaussian incompressible random field with the spectral tensor (2.95) constructed from the described expansion has the form n

uE N ,n .x/ D

.j /

n X N X  X iD1 j D1 rD1

®

p p

f .kijr /

pijr .kijr /

¯ .!ijr  ijr / cos. ijr / C .!ijr ijr / sin. ijr / ,

(2.98)

where ijr D 2kijr !ijr  x, and  ijr , ijr , i D 1, : : : , N ; j D 1, : : : , n ; r D .j /

1, : : : , n are mutually independent and independent of the family kijr , standard 3dimensional Gaussian random variables (with zero mean and unity covariance matrix).

2.7.2 Some technical details of FWM Simulation formula for a 3D isotropic incompressible random field uE .x/ with the spectral tensor (2.95) is constructed through the plane wave decomposition, without stratification, using the model (2.81), or with stratification, on the basis of (2.73) and (2.82), where the random processes vE.i/ .t /, i D 1, : : : , Ns are constructed by FWM (2.48). In our case, the 3D FWM based on the plane wave decomposition without a stratification has the form Ns 1 X .x/ D !i vE.i/ .!i  x/, (2.99) uE rnd Ns 1=2 Ns iD1 where !i , i D 1, : : : , Ns are mutually independent 3dimensional unit isotropic vectors, and vE.i/ .t /, i D 1, : : : , Ns are mutually independent 3-dimensional stationary Gaussian processes with independent identically distributed components. Each component of the process vE.i/ .t / has the spectral function F .k/ D E.k/. In the case of stratified models, the subdivision of the unit sphere appeared in (2.73) and (2.82) is constructed as described in the previous section. Thus the 3D FWM based on the plane wave decomposition with deterministic nodes (i. e., with (2.75)) is n X   X jj ,r j 1=2 n

uE det n .x/ D

.j /

j D1 rD1

4

j ,r vE.jr/ .x  j ,r /,

(2.100)

61

Section 2.7 Comparison of Fourier wavelet and randomized spectral models

where j ,r D .sin. j / cos.jr /, sin. j / sin.jr /, cos. j //, Z d  D 4j Œcos. j   =2/  cos. j C  =2/. jj ,r j D

j ,r

A hybrid stratified randomized FWM based on the plane wave decomposition (2.82) has the form n X   X jj ,r j 1=2 n

uE srm n .x/ D

.j /

j D1 rD1

4

!j ,r vE.jr/ .x  !jr /,

(2.101)

where !jr D .sin. Qj / cos.Qjr /, sin. Qj / sin.Qjr /, cos. Qi j //,   Qj D arccos .1  j / cos. j   =2/ C j cos. j C  =2/ , Qjr D jr C .jr  1=2/j , .j /

and j , jr , .j D 1, : : : , n ; r D 1, : : : , n / are mutually independent random numbers uniformly distributed on Œ0, 1. The stationary random processes .j / vE.jr/ .t / .j D 1, : : : , n ; r D 1, : : : , n /, appeared in (2.100) and (2.101) are mutually independent 3-dimensional stationary Gaussian processes with independent identically distributed components. Each component of the process vE.jr/ .t / has the spectral function F .k/ D E.k/ (see (2.95)) and is simulated by FWM (2.48). Since all three components of all these random processes have the same spectral function, in the sim./ . / ulation formulas one and the same set of functions Fm0 and Fm , m D m0 , : : : , m1 is used. Parameters of the Fourier wavelet model (2.48) are taken as b0 D b1 D 10; m0 D 0, m1 D 6. Now we give some details related to the simulation formulas (2.99)–(2.101). AsE srm sume we have to simulate a random field uE det n .x/ (or u n .x/ ) in a bounded domain .j /

D IR3 . We choose segments Œajr , bjr  .j D 1, : : : , n ; r D 1, : : : , n / so that x  j ,r 2 Œajr , bjr  8x 2 D. Then we simulate by the FW model (2.48) the processes .j /

vE.jr/ .t /, j D 1, : : : , n ; r D 1, : : : , n on a sufficiently fine grid on Œajr , bjr . The quantities vE.ij / .xi,j / are evaluated through a linear interpolation of the values vE.ij / obtained on this fine grid. The same can be done for the model (2.99). The Eulerian correlation functions (BLL .r/, BN N .r/, defined above are calculated through ensemble and spatial averaging, by FWM and RSM.

62

Chapter 2 Stochastic simulation of vector Gaussian random fields

2.7.3 Ensemble averaging In Figure 2.3 we plot the functions BLL .r/ and BN N .r/ for a Gaussian isotropic incompressible random field with the spectral tensor (2.95). The number of MC samples here and in all ensemble averagings was taken as 16,000.

2.7.4 Space averaging In this section we deal with the space averaging aiming at evaluating the Eulerian correlation functions on the basis of the ergodic property. Let us discuss this in more detail. Assume we need to evaluate the Eulerian correlation function BLL .r/ using only one sample of the random field uE .x/. Since uE .x/ is homogeneous, the random process .x; r/ D u1 .x Cr, 0, 0/u1 .x, 0, 0/, x 2 IR (r is fixed) is stationary, and by definition, h.x; r/i D BLL .r/. Assuming that this process is ergodic (this is true under quite general conditions; e. g., see [259]) P x we can evaluate h.x; r/i by a spatial averaging. .xi /, where xi , i D 1, : : : , nx is a set of points In particular, h.x; r/i ' n1x niD1 in IR. It is reasonable to choose the points so that the minimal distance between the points is larger than the characteristic correlation length of our random process .x; r/. These arguments can be extended to the case of points xi 2 IR3 which leads to the relations 1 np

BLL .r/ '

ny X

nx X

nz X

u1 .ix L C r, iy L, iz L/u1 .ix L, iy L, iz L/,

ix Dnx iy Dny iz Dnz

q

q

35

homogeneous z = 5 cm

20

30

0

homogeneous z0 = 25 cm

25

15

20

xd = 100 m

10

xd = 20 m

15

x = 50 m d

10

xd = 50 m

5

5

0

0 −2

10

x = 100 m d x = 20 m d

−2

−1

10

10

X/h

−1

10

X/h

Figure 2.3. The longitudinal (BLL ) and transversal (BN N ) correlation functions calculated by RSM (model (2.97) with n0 D 25, left panel), and FWM (model (2.99) with Ns D 25, right panel) models through ensemble averaging. Calculations: bold solid line; Explicit result (2.96): solid line. The number of MC samples in both cases was 16,000.

63

Section 2.8 Conclusions

1 BN N .r/ ' np

nx X

ny X

nz X

u2 .ix L C r, iy L, iz L/u2 .ix L, iy L, iz L/,

ix Dnx iy Dny iz Dnz

where L > 0 is the grid size, np D .2nx C 1/.2ny C 1/.2nz C 1/; in calculations we have taken nx D ny D nz D 7 (hence the number of points np D 3375), and L D 5. In Figures 2.4 and 2.5 the correlation functions BLL .r/ and BN N .r/ evaluated by RSM (2.97) for different values of n0 are presented. These results show that to reach the accuracy compared to that of the ensemble averaging calculations, (see Figure 2.3, left panel) even n0 D 1200 cannot be considered as satisfied. Close results were obtained by the model (2.98) for n  30 (i. e., Ns  1132). In Figure 2.6 we plot the functions BLL .r/ and BN N .r/ calculated by the use of models (2.99) (left panel) and (2.101) (right panel). The model (2.100) with n D 10 gave approximately the same results. The amount of statistics is np D 153 D 3375, so we see that the results are in a good agreement with the results of Figure 2.3 obtained through the ensemble averaging (note that the statistics in Figure 2.6 is about 4 times less than that of Figure 2.3). Thus we conclude that to calculate the Eulerian correlation functions BLL .r/ and BN N .r/ by the space averaging with one sample of the field uE .x/, the number of harmonics Ns in RSMs (2.97), (2.98) should be taken to be around several thousand, to attain the same accuracy as FWMs provide with several of hundreds of harmonics. (in both methods, statistics was 3375).

2.8

Conclusions

Simulation methods of homogeneous Gaussian random fields based on randomized spectral representations and Fourier wavelet decomposition were presented. Extensions of Fourier wavelet method (FWM) from scalar processes and isotropic fields to general vector fields were suggested. Convergence of the constructed Fourier wavelet models (in the sense of finite-dimensional distributions) under some general conditions on the spectral tensor were given. A comparative analysis of RSM and FWM was made by calculating the Eulerian statistical characteristics of a 3D isotropic incompressible random field through an ensemble and space averaging. The comparative analysis can be summarized as follows. 

In the case of ensemble averaging, in contrast to the 1-dimensional case, the complexity of the randomized spectral model (RSM) and 3D FWM are more or less compatible, because in FWM based on the plane-wave decomposition, the 1-dimensional processes are precalculated on a fine grid in advance, so that there is no need in the sophisticated management for random numbers choice. The best results are obtained when the number of 1-dimensional processes is about 10. Nevertheless, RSM seems to be preferable in this case, since its implementation is much easier.

64

Chapter 2 Stochastic simulation of vector Gaussian random fields

Q

Q

n

1

n

x = 200 m d

1

0.9

0.9

xd = 100 m

0.8

xd = 100 m

0.8

x = 50 m homogeneous z = 5 cm

0.6

0.6

x = 50 m d

0

0.5

0.5

xd = 20 m

0.4

xd = 20 m

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0

d

d

0.7

homogeneous z = 25 cm

x = 200 m

0.7

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0

0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

X/h

X/h

Figure 2.4. Eulerian longitudinal and transversal correlation functions calculated by RSM (2.97) with n0 D 100 (left panel) and n0 D 400 (right panel) obtained by space averaging. Bold solid line; calculations; solid line: explicit result. c

c

700

1000

xd = 50 m

900

600

x = 50 m d

800

500

x = 200 m

700

x = 100 m d

x = 20 m

d

d

600

400

x = 20 m d

x = 200 m

500

d

300 400

x = 100 m d

300

200

100

−3

−2

−1

10

0

100

0

10

homogeneous z = 1 cm

200

homogeneous z = 1 cm

−3

10

−2

10

X/h

−1

10

10

X/h

Figure 2.5. Eulerian longitudinal and transversal correlation functions calculated by RSM (2.97) with n0 D 800 (left panel) and n0 D 1200 (right panel) obtained by space averaging. Bold solid line: calculations; solid line: explicit result. C

C

homogeneous z = 1 cm

160

homogeneous z = 1 cm

160

0

0

140

x = 200 m

140

x = 200 m

d

d

120

120

x = 100 m

x = 100 m

d

100

d

100

x = 50 m

x = 50 m

d

80

d

80

x = 20 m

x = 20 m d

60

d

60

40

40

20

20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

X/h

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

X/h

Figure 2.6. Eulerian longitudinal and transversal correlation functions calculated by FWM (2.99) with N s D 200 (left panel) and (2.101) with n D 10 (right panel) obtained by space averaging. Bold solid line: calculations; solid line; explicit result.

65

Section 2.9 Appendices 

For space averaging, when evaluating Eulerian characteristics using one sample of the simulated random field, FWM shows higher efficiency because its samples have better ergodic properties. To get stable results by RSM, one needs a huge number (several thousands) of harmonics.

2.9

Appendices

2.9.1

Appendix A. Positive definiteness of the matrix B

Here we prove that the matrix B with entries (2.59) is positive definite, i. e., for any N-dimensional vector-rows c1 , : : : , cN 2 IRm Z ci bij cjT D ci B.ti  tj ; /cjT d   0 (A.1) Ai \Aj

is true for all integers N > 0 and values t1 , : : : , tN in IR, and measurable subsets A1 , : : : , AN in Sd 1 . N Let ¹D˛ ºM ˛D1 be a subdivision of the set A D [iD1 Ai (i. e., D˛ \Dˇ D ; if ˛ ¤ ˇ, and A D [M ˛D1 D˛ ) such that it is also a subdivision of Ai for any i D 1, : : : , N : i Ai D [N ˛j D1 D˛j . Let uE ˛ .t /, 1  ˛  M be a family of n-dimensional vector columns, mutually R independent zero mean Gaussian random processes with the correlation tensor D˛ B.t , / d . Let us construct, for each i D 1, : : : , N , a zero mean Gaussian ranP i dom process vEi .t / by putting vEi .t / D jND1 uE ˛j .t /. Then obviously Z hE vi .t /E vjT .t /i

D

Ai \Aj

B.ti  tj ; / d .

Introducing a random variable  D ci vEi .ti / (here we recall about the summation convention) yields h 2 i D hci vEi vEjT cjT i D ci bij cjT . which ensures (A.1).

2.9.2

Appendix B. Proof of Proposition 2.1

From the definition (2.48) we have for the random field u.F W / .1/ u.x/ D u.F W / .x/ C ıvm .x/ C 0

m1 X mDm0

.2/ .3/ ıvm .x/ C ıvm .x/, 1

(B.1)

66

Chapter 2 Stochastic simulation of vector Gaussian random fields

where

X

.1/ ıvm .x/ D 0

Fm./ .2m0 x C j / j , 0

j : jj b2m0 xcj>b0

X

.2/ ıvm .x/ D

Fm. / .2m x C j / mj ,

j : jj b2m xcj>b .3/ .x/ D ıvm 1

1

1 X

1 X

mDm1 C1

j D1

Fm. / .2m x C j / mj .

The random variables j , mj are mutually independent, hence the terms in the righthand side of (B.1) are also independent; therefore hu.x C r/u.x/i  hu.F W / .x C r/u.F W / .x/i D

.1/ .x hıvm 0

C

.1/ r/ıvm .x/i 0

C

m1 X

.2/ .2/ hıvm .x C r/ıvm .x/i

mDm0 .3/ .3/ .x C r/ıvm .x/i. (B.2) C hıvm 1 1

Let us estimate the terms in the right hand side. First, we have for the last term  1=2 .3/ .3/ .3/ 2 .3/ 2 jhıvm .x C r/ıv .x/ij  h.ıv .x C r// i h.ıv .x// i . (B.3) m1 m1 m1 1 Lemma 2.1. Assume that for some s > 0 Z 1 Is D F .k/.1 C jkj2 /s d k < 1. 1

Then a constant Bs depending only on s can be found such that Z 1 Bs2 .3/ 2 .x// i  F .k/.1 C jkj2 /s d k . h.ıvm 1 4.m1 C1/s 1

(B.4)

(B.5)

Proof. Note that 1 X

.3/ .x//2 i D h.ıvm 1

1 X

jFm. / .2m x C j /j2 .

(B.6)

mDm1 C1 j D1

Further, by (2.24) . /

Fm. / .2m x C j / D Gmj .x/ D

Z

1 1

e i2kx Q.k/ NO mj .k/ d k Z 1 D G.x C y/ 1

mj .y/dy,

67

Section 2.9 Appendices

R1

where G.x/ D

1 e

. / Fm .2m x

i2kx Q.k/ d k.

Consequently for fixed x, the quantities

C j / .m, j D : : : , 1, 0, 1, : : :/ are Fourier coefficients in the expansion of G.x C / with respect to the orthonormal system mj .m, j D : : : , 1, 0, 1, : : :/. Notice that the condition (B.4) can be formulated equivalently that the function G belongs to the space of Bessel potentials H2s .IR/, since G 2 H2s .IR/ means that 1=2

the function Q.k/.1 C jkj2 /s=2 is from L2 .IR/) and jjGjjH2s D Is (for more details see [243]). Further we use the fact that H2s .IR/ coincides with the Besov space s .IR/, and the norms in these spaces are equivalent (see [243]). On the other hand, B22 it is known (see [141]), that under quite general assumptions on the wavelet function .x/ (Meyer’s wavelet function satisfies this condition if the function .x/ is smooth s .IR/, 1  p, q  1, s > 0 can be s in Besov’s space Bpq enough) the norm jjf jjBpq R equivalently defined through the wavelet coefficients ˇmj D f .x/ mj .x/dx by jjf jj

s Bpq

D jjf jjLp C

² X  X 1 1  mD1

m.sC 12  p1 / p

q=p ³1=q

jˇmj j2

.

j D1

s , we conclude Since G 2 B22 1 X

1 X

mD1 j D1

2 jFm. / .2m x C j /j2 4ms  Bs2 jjG.x C /jjH s 2

2 2 for some Bs depending only on s. From this we get by jjG.x C /jjH s D jjGjj s D Is H2 2 that 1 X

1 X

jFm. / .2m x C j /j2

mDm1 C1 j D1

 

1 X

1 X

mDm1 C1 Bs2 4.m1 C1/s

j D1

jFm. / .2m x C j /j2

4ms 4.m1 C1/s

Is .

This completes the proof of Lemma 2.1. This implies due to (B.3) and (B.5) that the last term in the right-hand side of (B.2) can be estimated as follows: Z 1 Bs2 .3/ .3/ F .k/.1 C jkj2 /s d k . (B.7) jhıvm1 .x C r/ıvm1 .x/ij  .m C1/s 4 1 1 Now we turn to the estimation of the first two terms in the right-hand side of (B.2). We will use some results from the theory of Nikolski˘i–Besov spaces (e. g., see [151, 243]).

68

Chapter 2 Stochastic simulation of vector Gaussian random fields

A triple .r, j , l/ is called admissible if j 2 IN , l 2 IN0 and j > r  l. Here .j / IN0 D ¹0, 1, 2, : : :º and IN D ¹1, 2, : : :º. Let us denote by h g the j-th difference of g: .j / .j 1/ g./. h g./ D g. C h/  g./, : : : , h g./ D h h r .IR/ consists of all functions f such that For 1  p, q  1, r > 0 the space Bpq the norm r r , D jjf jjLp C jjf jjbpq jjf jjBpq

where Z jjf jj

r bpq

D

1



.j /

jjh f .l/ jjLp jhjrl

1

q

dh jhj

q1

1  q < 1,

,

.j /

r D sup jhj lr jjh f .l/ jjLp jjf jjbp1

(B.8) (B.9)

0 0,

(B.11)

l 2 IN0 ,

1  p < 1,

(B.12)

l 2 IN0 ,

1  p < 1,

(B.13)

where Wpl .IR/ is the Sobolev space, and X ,! Y for seminormed spaces X and Y means that X Y and there exists a constant c > 0 such that the inequality jjxjjY  cjjxjjX is fulfilled. r .IR/, r > 1=2. Then f is uniformly continuous, f 2 Lemma 2.2. Let fO 2 B11 L2 .IR/ and there exists a positive constant Cr depending only on r such that for all x 2 IR r . jxjr jf .x/j  Cr jjfOjjB11

(B.14)

Proof. The uniform continuity of f follows from the fact that fO 2 L1 .IR/ since r .IR/ L .IR/. Then, for a positive " 2 .0, r  1=2/ in view of (B.10) and (B.11) B11 1 r1=2

r B11 .IR/ ,! B21

r1=2"

.IR/ ,! B22

.IR/.

(B.15)

69

Section 2.9 Appendices

From this we get by (B.13) that fO 2 L2 .IR/, hence f 2 L2 .IR/. Let l D brc be the integer part of r. From Z 1 .lC2/ O f .k/ D h e i 2 kx .e i 2 hx  1/lC2 f .x/ dx (B.16) 1

and f 2 L2 .IR/ it follows by the inverse Fourier transform Z 1 .lC2/ O f .k/ d k. e i 2 kx h f .x/.e i 2 hx  1/lC2 D 1

(B.17)

Taking the absolut values and dividing this equation by jhjr we then take the supremum over h 2 IR. This yields jxjr jf .x/jCr0  sup jhj r jjh

.lC2/

h2IR

where

´ Cr0

D sup

t2IR

je i 2 t  1jlC2 jt jr

fOjjL1 ,

(B.18)

.

(B.19)

μ

.lC2/ O f jjL1  Since the triple .r, l C 2, 0/ is admissible, and suph2IR jhj r jjh 00 00 O r Cr jjf jjB11 for some Cr depending only on r, the proof of Lemma 2.2 is complete. .1/

.1/

To get an estimation for hıvm0 .x C r/ıvm0 .x/i we estimate the quantity X .1/ 2 .x// i D jFm./ .2m0 x C j /j2 . h.ıvm 0 0

(B.20)

j : jj b2m0 xcj>b0 ./

We first estimate each term of this sum. The function Fm .y/ has the Fourier transNO (see (2.23)). form 2m=2 Q.2m k/.k/ O Hence if  is smooth enough and if the function Q.k/ belongs to the Nikolski˘ı–

ON Besov space B .IR/,  > 1=2, then 2m=2 Q.2m k/.k/ 2 B .IR/. Then from 11

11

(B.14) we get jFm./ .y/j 

C m=2 NO  D C m ., Q/ , jj2 Q.2m /jj B11

jyj jyj

(B.21)

where C m ., Q/ is a constant depending on , m and functions  and Q. Note that from (B.20) and (B.21) we get .1/ .x//2 i  h.ıvm 0

0 Cml

2 1 . b0 The same arguments are used to estimate the terms in the right hand side of (B.2) .2/ related with ıvm . The proof of Proposition 2 is complete.

Chapter 3

Stochastic Lagrangian models of turbulent flows: Relative dispersion of a pair of fluid particles Stochastic Lagrangian models of relative motion of two fluid particles in one and three dimensions for locally isotropic incompressible turbulent flow are presented. A principle of consistency of statistics between the Eulerian and Lagrangian velocity fields for general random forcing models of a diffusion type is proposed. This enables us to analyze and improve some well known models. An analog of the well-mixed condition for the relative dispersion is proposed. An explicit form of an 1-order consistency model is given. While there is a series of applications (both in theory and applied turbulence) of the relative motion of two particles, a relation to the problem of concentration fluctuations reflect our motivation by the problem of pollutant scattering in the turbulent atmosphere.

3.1 Introduction There are two main approaches in stochastic modeling of turbulent motion of fluid particles. The first one uses models of Eulerian velocity field UE .x, t /, for instance, randomized Monte Carlo models [191, 197], or models based on numerical methods for solving the Navier–Stokes equation [24, 160]. With such models, one simulates an ensemble of Lagrangian stochastic trajectories and calculates the desired statistical characteristics. A second approach uses different approximations of the Lagrangian velocities of a set of particles. The first model of this type was introduced by G. Taylor [235] in 1921, and was then developed by many authors (see e. g., [91, 108, 133, 157, 214, 236]). The first approach is rigorous, but it requires a lot of computer time. The second is much more effective, but its correct utilization needs special justifications. Both approaches are useful for modeling the pollutant scattering in turbulent flows, and especially for calculating the concentration fluctuations (see e. g., [144,146,214]). Most developed are the 1-particle Lagrangian models which allow the calculation of only the mean concentration. To find the concentration fluctuations, or, more generally, the covariance of the concentration, it is necessary to have 2-particle Lagrangian models. Unfortunately, such models are not well developed. The motion of two particles can often be regarded as a motion of one particle plus the relative motion of the second particle with respect to the first one. Compared to the absolute diffusion of two particles, it is believed that the relative diffusion can be treated, due to the Kolmogorov similarity hypotheses, using a simpler and universal

71

Section 3.1 Introduction

description. Therefore, models of relative diffusion attract much attention (see e. g., [44, 45, 108, 155, 236]). We mention also 2-particle Gaussian models [123, 214] and the model [20]. We give here two examples to motivate why the relative diffusion is of particular interest. Assume that we describe a pollutant transport in a homogeneous velocity field, and c.x, t /, the instantaneous concentration of the pollutant is also statistically homogeneous. Then Bc .r, t /, the covariance of the concentration Bc .r, t / D hc.x, t / c.x C r, t /i is represented as (see e. g., [135]) Bc .r, t / D

Z p.r, t j r0 /Bc .r0 , 0/ dr0 .

(3.1)

Here p.r, t j r0 / is the probability density function (p. d. f.) p.r, t j r0 / D hı.r  rL .t , r0 /i, where rL .t , r0 / is the instantaneous separation (at the time instant t ) between the particles initially separated by r0 . In the derivation of .3.1/ it is supposed that the initial concentration field c.x, 0/ is statistically independent of the velocity field. Thus, to evaluate the covariance of concentration, it is necessary (and sufficient, in this case) to know the statistics of the relative motion of the two particles. The second example comes from the Smolouchovsky equation governing the coagulation of particles in homogeneous turbulent flows [191], where the coagulation rates depend on the relative velocity between two colliding particles. In this chapter we deal with the problem of the relative motion of two particles. We suggest different stochastic models (3D and quasi-1-dimensional) which are consistent in a sense with the Eulerian velocity field. Let us explain this point in more detail. In statistical fluid mechanics [146] the Eulerian turbulent velocity field is considered as a 3D random field UE .x, t / depending on a spatial coordinate x and time t . By fluid particles we mean mathematical points moving with the local velocity of the flow. The Lagrangian trajectory X.t , x0 /, t  0 of a fluid particle originating at position x0 at time t D 0 is found from the equation of motion: @X D UE .X , t / @t

(3.2)

X.t D 0, x0 / D x0 .

(3.3)

subject to the initial condition

The Lagrangian velocity V .t , x0 / is related to the Eulerian velocity UE .x, t / by V .t , x0 / D UE .X.t , x0 /, t /.

(3.4)

72

Chapter 3 Stochastic Lagrangian models and relative prodispersion

Equation .3.2/ can be integrated numerically in time; .3.4/ means that the instantaneous particle velocity is the same as the fluid velocity at the instantaneous particle position. Each Eulerian quantity f .x, t / can be related to the Lagrangian quantity by F .x0 , t / D f .X.x0 , t /, t /,

x0 2 R 3 ,

t  0.

(3.5)

A Lagrangian description allows us to analyze directly the motion of material fluid elements. Unfortunately, the Lagrangian statistics are extremely difficult to measure accurately. Numerical solution of .3.2/ based on different representations of the random field UE involves nontrivial difficulties which must be overcome before we can have confidence in the accuracy of the result. The second approach we mentioned above is based on direct approximations of the stochastic Lagrangian velocity V .t , x0 /, say VO .t , x0 /. The relevant approximate Lagrangian trajectory is then obtained from the ordinary stochastic differential equation d XO D VO .t , x0 / dt

(3.6)

O 0 , 0/ D x0 . Note that this is generally a nonlinear equation since the random with X.x process VO .t , x0 / itself depends on the random position XO .t /. Most frequently, models of the type .3.6/ are written in the form of stochastic differential equations (SDE) of the Ito type. These equations are much easier to solve numerically than the general equations of fluid particle motion (3.2) in random fields. Numerical methods for solving SDE are well developed (see e. g., [233]. The models (3.2) and (3.6) define one and the same flow if VO .t , x0 / D V .t , x0 / for all Lagrangian trajectories. It would be an ideal approximation if we could find V .t , x0 /. Of course, it is impossible to do even for the simplest flows. However, when constructing models of the type (3.6), it is reasonable to use some relations between the statistics of the Eulerian velocity field UE .X , t / and of the 6-dimensional Lagrangian field Y .t , x0 / D .X.t , x0 , V .t , x0 //. For instance, in the case of incompressible flows there is an integral relation between the probability density functions of UE .x, t / and Y .t , x0 / (see [153]; the more general situation is treated in [154]). In addition to this kind of consistency, we will use also some well-known properties of the stationary local isotropic incompressible turbulence such as isotropy and the Kolmogorov similarity hypotheses, which assume that there exists a parameter "N (called the mean rate of dissipation of kinetic energy) such that the statistical characteristics of the increments of the Eulerian velocity after normalizing by a quantity v become universal functions (for all kinds of fully developed turbulence) of a nondimensional time t = and nondimensional spatial distance r= . Here is Kol-

73

Section 3.2 Criticism of 2-particle models

mogorov’s spatial microscale which is related to the velocity microscale v and the time microscale  by

D . 3 =N"/1=4 , v D . "N/1=4 ,  D =v , where  is the kinematic viscosity of the flow.

3.2

Criticism of 2-particle models

We start with a critical analysis of some known 2-particle stochastic models. Assumably, the first 2-particle model appeared in the early 60s, in [129,152]. In [129], it was assumed that the random process in the 6-dimensional phase space .r, v/ (r is the separation vector between the two particles, and v is the relative velocity) is Markovian. Using then the similarity and isotropy assumptions, it is possible to derive an equation governing the p. d. f. [17]. Novikov [152] assumed that the motion of two particles is governed by two independent Langevin equations: dX .i/ .t / D u.i/ , dt p 1 du.i/ D  u.i/ dt C u 2=TL d W .i/ , TL

(3.7) i D 1, 2, 3.

(3.8)

.i/

Here TL is the Lagrangian time scale, u2 is the variance of u1 , and W .i/ , i D 1, 2, 3 are independent standard Wiener processes. Subtracting the two equations yields dr D v, dt

dv D 

p 1 vdt C 2u 1=TL d W , TL

(3.9)

where r D X .2/ X .1/ , v D u.2/ u.1/ . Note that the p. d. f. of this equation coincides with the p. d. f. of the Markovian model [129]. By construction, this model correctly describes the motion of particles only for large separations, where the correlation between the velocities vanishes. A detailed analysis of this model is given in [17]. Models which take into account the statistical structure of the velocity in the inertial subrange were developed in [44, 45, 108, 155, 236]. Note that the 1-dimensional model [44] and its modification [236] have no adequate physical interpretation (see the introductory part in Section 3.3). However, quasi-1-dimensional models which describe the evolution of the distance between two fluid particles in 3D space have clear physical meaning. The motion of two particles in the framework of a quasi-1-dimensional model [45] is governed by the equation d 1=2 D RE ./ U.t / dt

(3.10)

74

Chapter 3 Stochastic Lagrangian models and relative prodispersion

with the initial condition: .0/ D 0 . Here  D .t , 0 / is the distance between the particles at the time instant t , and  1=3 2 , (3.11) RE ./ D 2 C L2 L is the integral spatial scale of the Eulerian velocity field, and U.t / is the Uhlenbeck– Ornstein process (UO): p 1 (3.12) d U D  Udt C U 2=TL d W . TL Obviously,

d (3.13) .t D 0, j0 j/ D uE .j0 j/ dt for all initial distances 0 , where uE .j0 j/ is the Eulerian longitudinal relative velocity. 1=2 The coefficient RE ./ in (3.10) was introduced to ensure that ²

³2 ˛ ˝ d .t D 0, j0 j/ D u2E .j0 j/ D U2 RE .j0 j/. dt

(3.14)

Note that at large separations the model (3.9) behaves like the model (3.7) (RE ./ ! 1 as  ! 1), while in the inertial subrange the model should have reflected the Kolmogorov law, RE ./  2=3 . However it appears [108] that the particles, starting at a distance from the inertial subrange, will collide with probability 1=2 at a point after time of the order of .20 =N"/1=3 , which is much less than TL . This makes the Durbin model questionable in practical usage. A criticism of Durbin’s model was given in [46]. Thomson [236] has modified Durbin’s model by adding an additional 1=2 term dRd . / . The same could be done with the quasi-1-dimensional model: d 1=2 D RQ E ./ U.t /, dt

1=2 1=2 d RQ E U 2 dt C d W .t / C dt , dU D  TL TL d where RQ E ./ D 2 2



2 2 C L2

(3.15) (3.16)

1=3 ,

and  2 is the variance of a component of the velocity. The shortcoming of the Durbin model mentioned above seems to disappear in this model since the new term increases the divergence between the two particles as  tends to zero. However, we will see later on that this modified model contradicts the integral relation (3.31) which is a

75

Section 3.2 Criticism of 2-particle models

consequence of incompressibility. We will further modify this model to provide some improvement ensuring a relevant consistency. An interesting approach was proposed by Novikov in [155] where the Lagrangian relative diffusion model consistent with some features of the inertial subrange was constructed. In the next section we will use this approach; therefore, let us give some details. The general 3D model has the form dvi D gi .t , r, v/ dt C di , dri D vi dt ,

i D 1, 2, 3,

(3.17)

where di stands for the random forcing, and r.0/ D r0 , v.0/ D ,  being a random vector with p. d. f hı.v  /i D pE .vj r0 /. Here pE .vj r0 / is the p. d. f. of the Eulerian relative velocity between two points separated by r0 . Then the Lagrangian probability density pL .t , v, rjr0 / D hı.v  v.t , r0 // ı.r  r.t , r0 //i for the relative velocity and distance is governed by the master equation [24] @ @pL @pL C .gi pL / C vi @t @ri @vi Z D pL .t , v  uj r0 / q.uj v  u, r/ du  pL .t , vjr0 / m0 .v, r/, (3.18) where

Z m0 .v, r/ D

q.uj v, r/ du,

and q.uj v, r/ is the transition probability per unit time of the jump (from v to v C u). Here and all through the chapter we will use (if not specially noted) the summation convention taking a sum over the repeated subscript. The Kramers–Moyal expansion of (3.18) gives (see e. g., [248]) 1 X @ @n @pL .1/n @pL .n/ C .ai pL / D . mi1 ,:::,in pL /, C vi @t @ri @vi nŠ @v : : : @v i1 in nD2 .1/

where ai D gi C mi , and .n/ mi1 ,:::,in .v, r/

Z D

ui1 : : : uin q.uj v, r/ du.

(3.19)

76

Chapter 3 Stochastic Lagrangian models and relative prodispersion

In the diffusion approximation, when (3.17) is a stochastic differential equation of the Ito type dvi D gi .t , r, v/ dt C bij d W ,

dri D vi dt ,

i D 1, 2, 3,

equation (3.19) reduces to @ 1 @2 @pL @pL .2/ C .ai pL / D . mi,j pL /, C vi @t @ri @vi 2 @vi @vj

(3.20)

(3.21)

.2/

where the matrix .bij /3i,j D1 is the square root of the matrix .mij /3i,j D1 . Under the incompressibility hypothesis, the following relation between pL and pE , the Eulerian probability density function, can be derived [155]: Z pL .t , v, rj r0 / dr0 . (3.22) pE .t , vj r/ D Integrating (3.21) over r0 and using (2.14) one gets the same equation for pE : @ 1 @2 @pE @pE .2/ C .ai pE / D . mi,j pE /. C vi @t @ri @vi 2 @vi @vj

(3.23)

From this it is not difficult to derive equations for the statistical moments. Thus we conclude that this attempt to construct a model of type (3.17), specifying .2/ the three first moments of pE , leads to some constraints on ai and mij . This promising approach needs detailing of the strategy of finding the functions gi and the statistical structure of the random forcing di . We do not know any detailed model satisfying the constraints mentioned above. In the last section of the chapter we suggest (in the framework of diffusion approximation) a model which satisfies (3.23). Note that if one wishes to include into a model important features of the turbulence, for instance, its structure in the inertial subrange, then it is necessary to give a formulation of the Lagrangian analog of these details. For example, in accordance with the well-known second Kolmogorov similarity hypothesis [146] there exists a universal stationary stochastic process E . / defined by uE .r, t / D E . /, (3.24) .N"jrj/1=3 t .r 2 =N"/1=3

D ,

(3.25)

where uE .r, t / is the relative longitudinal Eulerian velocity between two points whose separation is r, at the time instant t ;  is a nondimensional time. In [108], an analog of (3.24, (3.25) in Lagrangian formulation was supposed: d 1 (3.26) D L . /, .N"/1=3 dt 1 .2 =N"/1=3

dt D d  ,

(3.27)

Section 3.3 The quasi-1-dimensional Lagrangian model of relative dispersion

77

where  is the distance between the Lagrangian fluid particles, and L . / is assumed to be a universal stationary process. Note that the model (3.26, 3.27) d D .N"/1=3 L . / dt

(3.28)

looks like the Durbin model [45] in the inertial subrange. The difference lies in the temporal variable: in (3.9), the UO process depends on the physical time t while in (3.28) the stationary random process L . / is a function of a nondimensional time which is related to t through the local relation (3.27). This is a crucial feature of our model which overcomes the drawback mentioned above. Note that in the model (3.26, 3.27) we are free to choose the stationary stochastic process L . /. In Section 3.3 we give a special choice of this process which makes the model consistent with the Eulerian velocity statistics. In conclusion we mention the Gaussian 2-particle models [123, 213, 214] and the model [20] adjusted to the viscous subrange. In the inertial subrange the velocity distribution is not Gaussian; therefore, the Gaussian models [123, 213] could be applied to describe the dispersion of particles by energy containing subranges. Finally, there are also approaches intermediate between the Lagrangian and Eulerian descriptions (see e. g., [91]) which are beyond the scope of this section.

3.3

The quasi-1-dimensional Lagrangian model of relative dispersion

Note that there are 1-dimensional models which deal with the description of one of the components of the separation vector r.t / (e. g., [44,214]); these models do not adequately describe important features of the real turbulence. As an example, note that there are events where the distance between particles is large (and hence the relative velocity should also be large) while a projection of the separation vector might nevertheless be small which implies that the 1-dimensional model predicts a small relative velocity. We call a model of relative dispersion quasi-1-dimensional if it describes the evolution of the distance .t / between two fluid particles in the space R3 . Since the quasi-1-dimensional model describes the motion of two particles in 3D, it is reasonable to require that this model be consistent with the integral relation (3.22) in a sense given in the next subsection. In this section we give an approach to make a choice of L . / in (3.26, 3.27) which ensures that an important integral relation between the Eulerian and Lagrangian statistics mentioned above is valid. In the next subsection we derive this relation which is a consequence (not a straightforward one) of the Novikov integral relation (3.22).

78

Chapter 3 Stochastic Lagrangian models and relative prodispersion

3.3.1 Quasi-1-dimensional analog of formula (2.14a) Assumptions 3.1. We assume that UE .x, t / is a stationary locally isotropic 3D incompressible velocity field [146]. We use the following notations: X.t , x0 / is the Lagrangian coordinate, and V .t , x0 / is the Lagrangian velocity of a fluid particle at the time instant t , which started at the point x0 , i. e., @X V D (3.29) D UE .X , t /, X.0/ D x0 . @t The relative Eulerian velocity is uE .r, t / D UE .x C r, t /  UE .x, t / between two fixed points x and x C r. The distribution density function of uE is pE .uj r/ D hı.u  uE .r, t //i.

(3.30)

The Lagrangian relative velocity is vL .t , r0 / D V .t , x0 C r0 /  V .t , x0 / between two fluid particles which were separated initially by the vector r0 , and the p. d. f. of vL : pL .t , v, rj r0 / D hı.v  vL .t , r0 // ı.r  rL .t , r0 //i, where rL .t , r0 / D X.t , x0 C r0 /  X.t , x0 / is the separation vector. Remark 3.1. Note that the field uE .r, t / depends on x, while vL depends on x0 . However, since UE is locally isotropic, the densities pE and pL do not depend on x and x0 , respectively. We will also use a longitudinal Eulerian velocity which is defined as the scalar product k uE ., t / D .uE .!, t /, !/, where  D jrj is the distance between the two points, and ! D k vector. The p. d. f. of uE ., t / is denoted as k

r jrj

is a unit directional

k

pE .uj / D hı.u  uE ., t //i. The longitudinal Lagrangian velocity is defined by the following scalar product k

vL .t , 0 / D

.vL .t , 0 !0 /, rL .t , 0 !0 // , jrL .t0 , 0 !0 /j

Section 3.3 The quasi-1-dimensional Lagrangian model of relative dispersion

79

and the distance between two fluid particles is L .t , 0 / D jrL .t , 0 !0 /j, where 0 D jr0 j, and !0 D r0 =jr0 j. k The density of vL is k

k

pL .t , v, j 0 / D hı.v  vL .t , 0 // ı.  L .t , 0 //i. k

k

By definition, the instantaneous velocities uE and vL depend on ! and !0 , respectively, while the p. d. f.’s pE and pL are independent of these arguments. We now present the following statement which is an analog of (3.22) in the quasi1-dimensional case. Proposition 3.1. Under the assumptions given above, the following integral relation holds: Z 1 k k 20 pL .t , v, j 0 / d0 D 2 pE .vj /. (3.31) 0

N , x0 C 0 !0 / startN , x0 /, X.t Proof. Let us consider two Lagrangian trajectories X.t ing at x0 and x0 C 0 !0 , respectively, where !0 is a unit isotropic random vector independent of the velocity field, and let us introduce a p. d. f. by pQL .t , v, rj 0 / D hı.v  vQL .t , 0 // ı.r  rQL .t , 0 //i,

(3.32)

where vQL .t , 0 / is the relative velocity, and rQL .t , 0 / is the separation between the two fluid particles. By the formula of conditional expectations, the function pQL .t , v, rj 0 / is related to pL .t , v, rj r0 / by Z pQL .t , v, rj 0 / D pL .t , v, rj0 !0 / d .!0 /, (3.33) 

where d .!0 / D .sin d /.d'=4/ is the normalized surface measure on the unit sphere . We introduce the density fL0 .t , v, , !j 0 /  rQL .t , 0 / D ı.v  vQL .t , 0 // ı.  jrQL .t , 0 /j/ ı !  (3.34) jrQL .t , 0 /j and the conditional density fL .t , v, j !, 0 / D R

fL0 .t , v, , !j 0 / . fL0 .t , v, , !j 0 / dv d

(3.35)

80

Chapter 3 Stochastic Lagrangian models and relative prodispersion

The integral in (3.35) is equal to 1=4, since the integrand is a p. d. f. of the vector rQL .t , 0 /=jrQL .t , 0 /j, which is isotropic. Simple geometric arguments show that fL0 .t , v, , !j 0 / D 2 pQL .t , v, !j 0 /.

(3.36)

From (3.35) and (3.36) we get fL .t , v, j !, 0 / D 42 pQL .t , v, !j 0 /. By the definition, k pL .t , v3 , j

(3.37)

“ 0 / D

fL .t , v, j e3 , 0 / dv1 dv2 ,

(3.38)

where v D .v1 , v2 , v3 / and e3 D .0, 0, 1/. Multiplying the last equality by 20 and integrating over 0 we get, using (3.37) and (3.33), Z 1 k 20 pL .t , v3 , j 0 / d0 0 Z 1 Z 1 Z 1 D 20 d0 dv1 dv2 42 pQL .t , v, e3 j 0 / 0 1 1 Z 1 Z 1 Z 1 Z 2 2 D 0 d0 dv1 dv2 4 pL .t , v, e3 j 0 !0 / d 0 .!0 / 0 1 1   Z 1 Z 1 Z 1 Z 2 2 D dv1 dv2 4 0 d0 pL .t , v, e3 j 0 !0 / d 0 1 1 0  Z Z 1 Z 1 D 2 dv1 dv2 pL .t , v, e3 j r0 / dr0 / 1 1 R3 Z 1 Z 1 k D 2 dv1 dv2 pE .vj e3 / D 2 pE .v3 j /. (3.39) 1

1

This completes the proof. This result will be used in Section 3.3.3 when constructing the stochastic process L . / in (3.26).

3.3.2 Models with a finite-order consistency Let us consider the following system of stochastic differential equations modeling the evolution of the distance L .t , 0 / between two fluid particles in 3D space: OL d k D vOL , dt k O L , vO k /dt C b. O L , vO k / d W .t /. d vOL D a. L L

(3.40) (3.41)

81

Section 3.3 The quasi-1-dimensional Lagrangian model of relative dispersion

O L .t , 0 / and vO k .t , 0 / for the solution to (3.40, 3.41) satWe will use the notation  L isfying the initial conditions O L .t D 0, 0 / D 0 , vO k .t D 0, 0 / D 0 ,  L

(3.42)

k

where 0 is a random variable with p. d. f. hı.v  0 /i equal to pE .vj 0 /. Our general objective is to determine the coefficients in (3.41) in such a way that the density k

k

k

O .t , 0 //i pOL .t , v, j 0 / D hı.v  vOL .t , 0 // ı.   L k

when substituting into (3.31) (replacing pL ) will satisfy this integral relation. If so, then, as in the case of derivation of (3.23), we arrive at the following PDE: v

k  @Œ2 pE .vj / 1 @2  2 2 @ k k  b ., v/p .vj / . C Œ2 a., v/pE .vj / D E @ @v 2 @v 2 (3.43)

Z

Let n ./ D

k

v n pE .vj / dv,

n D 0, 1, : : : .

It is obvious that 0 D 1, 1 D 0, 2 ./ D DLL ./,

(3.44)

and in the inertial subrange 2 ./ D C.N"/2=3 ,

4 3 ./ D  ". N 5

(3.45)

The function DLL and the constant C are defined below in Section 3.5.3. Multiplying (3.43) by v n and integrating over v yields Z 1 d k v n1 a., v/pE .vj / dv Œ2 nC1 ./  n2 d 1 Z n.n  1/ 2 1 n2 2 k D v b ., v/pE .vj / dv (3.46)  2 1 provided the functions k

v n a., v/pE .vj /, v n

@ 2 k k Œb ., v/pE .vj /, v n1 b 2 ., v/pE .vj / @v

tend to zero as jvj ! 1. We say that the model (3.40), (3.41) satisfies the m-order consistency condition if (3.46) is true for all n  m. Note that the 1-order consistency implies that (3.43) is fulfilled.

82

Chapter 3 Stochastic Lagrangian models and relative prodispersion

Let us analyze the consistency property of the models (3.9) and (3.14). For simplicO and vOL , respectively. Let us express the ity, we use the notation  and v instead of  Durbin quasi-1-dimensional model (3.9) and the Thomson modified model (3.14) in the general form d D R1=2 ./U.t /, dt   1=2

dR1=2 2 U C d W .t /. dt C dU D  TL d TL

(3.47)

For the sake of simplicity we use here the notation R./ D RQ E ./;  is a dimensionless parameter. From the Ito formula we arrive at the standard form of the type (3.40), (3.41) with

  dR./ v 2R./ 1=2 v 2 dR./ C . (3.48)  , b., v/ D a., v/ D 2R./ d TL 2 d TL In the inertial subrange R./  2=3 , and we get from (3.45), (3.48), and (3.46) with n D 1 that the first-order consistency condition is satisfied only if  D 7. Hence the Durbin model ( D 0) and the Thomson modification ( D 1) do not satisfy the first-order consistency condition. Note that there is no value of  providing the second-order consistency. We now suggest a model for the inertial subrange with a second-order consistency: d D R1=2 ./.t /, dt   1=2

dR1=2 ./ 2  d W .t /, C dt C ˛ d D   ./ d  ./

(3.49)

where  ./ D ˇ.2 =N"/1=3 , and R./ D C.N"/2=3 ; ˛, ˇ and  are nondimensional constants. From (3.49) we get, using the Ito formula, the standard form (3.41) with a., v/ D

v2 C  "N2=3 "N1=3 v C ,  3 ˇ2=3 31=3

b., v/ D ˛Œ2C "=ˇ N 1=2 .

(3.50)

If we suppose that pE .vj / satisfies the conditions at the infinity mentioned above then the zero-order consistency condition is fulfilled automatically. To provide the first-order and second-order consistency we found the constraints  D 7, ˛ 2 D 1 

14ˇ . 15C

Here we have one free parameter ˇ satisfying the inequality 0 < ˇ < 15C =14.

(3.51)

Section 3.3 The quasi-1-dimensional Lagrangian model of relative dispersion

83

Note that the last model reflects properly the well known Kolmogorov second similarity hypothesis (compare the functions b in (3.50) and (3.48)). When  L, then the motion of two particles can be considered as independent; hence, it is reasonable to assume that, due to isotropy,  d 2 (3.52) D 2 2 , dt where  2 is the variance of a component of the Eulerian velocity. Note that the Durbin model and the Thomson modification both possess this property. It is not difficult to generalize the model (3.49) in such a way that (3.52) will be satisfied. Indeed, let us take the constants ˛, ˇ and  as mentioned above (see (3.23)), and let us choose  ./ as 1=3

.L/2 . (3.53)  ./ D ˇ .2 C L2 /N" Define a function .x/ D 1 C

1˛ .1  exp.x 2 //, x  0. ˛

The modified model d D R1=2 ./.t /, dt    1=2

dR1=2  2  d W .t /, C dt C ˛ d D   ./ d L  ./

(3.54)

satisfies the desired condition (3.52) if in (3.53) the relation  2 3=2 2 1 "N D C L is taken into account. In the next subsection we suggest a model with an 1-order consistency.

3.3.3

Explicit form of the model (3.26, 3.27)

We will work here with the model OL d O L /1=3 L . /, D .N" dt  "N 1=3 d , D O2 dt  L d L . / D a.L / d  C  .L / d W . /.

(3.55) (3.56) (3.57)

84

Chapter 3 Stochastic Lagrangian models and relative prodispersion

Here W . / is the standard Wiener process, and a,  are some functions to be defined. OL O L to distinguish between the model distance  Note that we use here the notation  and the true distance L . To complete the construction of the model it remains to give the explicit form of a and  . It is natural to postulate that the p. d. f. pOL .t , v, j 0 / of the vector 

O L .t , 0 / d O L .t , 0 / ,  dt

satisfies, as the true density pL .t , v, j 0 / does, the equality Z 1 k k 20 pOL .t , v, j 0 / d0 D 2 pE .vj /.

(3.58)

0

Thus we can use the information about the Eulerian velocity field, namely, we suppose that pE .vj / is given. Starting from (3.55)–(3.58), we derive the desired coefficients a and b. First, we rewrite the system, using a random change of variables. Let WQ .t / be a standard Wiener process, and let QL .t / D L . .t //. Then, using the relation r d d W . / D d WQ .t /, dt from (3.55)–(3.58) we get OL d O L /1=3 QL .t /, D .N" dt !1=3 "N Q d L D a.QL / dt C O2 

(3.59) "N O2 

!1=6  .QL / d WQ .t /.

(3.60)

O L .t , 0 //i, fOL .t , , j 0 / D hı.  QL .t , 0 // ı.  

(3.61)

L

L

We introduce a p. d. f.:

Q L .t , 0 / is the solution to (3.59), (3.60) with the initial where the pair QL .t , 0 /,  conditions O L .t D 0, 0 / D 0 , QL .t D 0, 0 / D E ,  and E is a random variable with the p. d. f.:   k fE ./ D .N"0 /1=3 pE .N"0 /1=3 j 0 ,

1 <  < 1.

(3.62)

Note that fE does not depend on 0 in the inertial subrange, in accordance with the second Kolmogorov hypothesis.

Section 3.3 The quasi-1-dimensional Lagrangian model of relative dispersion

85

The function (3.61) satisfies the Kolmogorov–Fokker–Planck equation

 1=3  i @ @ h "N @fOL OL a./ f C .N"/1=3  fOL C @t @ @ 2  i 1 "N 1=3 @2 h 2 OL , D  ./ f 2 2 @ 2

(3.63)

with fOL .t D 0, , j 0 / D ı.  0 /fE ./. We also assume that the following condition is satisfied: ˇ

ˇ 1 @ 2 O D0 (3.64) . fL /  afOL ˇˇ 2 @ D˙1 and that the problem (3.63), (3.64) has a unique solution. The equality (3.58) can be rewritten in fOL-notation, using (3.62): Z 1 20 fOL .t , , j 0 / d0 D 2 fE ./. (3.65) 0

Multiplying (3.63) by 20 and integrating over 0 we get 

 1=3  @ @  2 "N a./f ./  .N"/1=3 fE ./ C 2 E @ @ 2  2 "N 1=3 @2 2 D Œ ./fE ./, 2 2 @ 2 which can be simplified to 1 d2 2 d 7 Œ ./fE ./ D Œa./fE ./ C fE ./. 2 2 d d 3 Let

Z F ./ D

and let ˆ./ D

(3.66)



1

fE . / d ,

1 d 2 . fE .//  a./fE ./. 2 d

(3.67)

(3.68)

Integrating (3.66) we get 7 ˆ./ D ˆ.1/ C F ./. 3

(3.69)

The conditions (3.64) imply that ˆ.1/ D 0; hence we come to the system 7 1 d 2 Œ ./fE ./  a./fE ./ D F ./. 2 d 3

(3.70)

86

Chapter 3 Stochastic Lagrangian models and relative prodispersion

We now choose a function  and make the following assumptions: (a) The SDE d . / D . / d C d W . /

(3.71)

has a unique solution defined for all the time instances, and (b)

Z If 

1

² Z jF .t /j exp 2

0

³

t

. / d 

dt < 1.

(3.72)

0

We will also assume that a./ D ./ 2 ./.

(3.73)

Let

1 ./ D  2 ./fE ./. 2 Then from (3.70), (3.73), and (3.74) we get d 7 D 2././ C F ./. d 3

(3.74)

(3.75)

We seek a positive solution to (3.75), since, from (3.74), we see that   0. We assume that 7 .0/ D 0 > If . (3.76) 3 The explicit solution to (3.75) is ² Z  ³

² ³ Z  Z 7  ./ D exp 2 . / d 0 C d F . / exp  2 . / d  , (3.77) 3 0 0 0 which is positive, in view of (3.76). Proposition 3.2. Under the assumptions (a), (b), (3.76) and (3.73), the solution to (3.57) with (3.78) a./ D 2././=fE ./ and  ./ D ¹2./=fE ./º1=2 ,

(3.79)

where ./ is given by (3.77), is unique for each initial condition L .0/ D 0 and is defined for all the time instances  . The solution to (3.57) with these coefficients is ergodic. Proof. We use the following random transformation of time: fE .. // d D , d 2.. //

(3.80)

Section 3.3 The quasi-1-dimensional Lagrangian model of relative dispersion

87

Q / D .. //. From (3.57) we get by the Ito formula that and introduce a function . " #1=2 Q / Q Q 2a. /. 2. / Q /D Q d . d WQ . /, d C  ./ Q Q fE ./ fE ./

(3.81)

where WQ . / is a standard Wiener process. Substituting the coefficients (3.78), (3.79) Q Hence the random transformation (3.80) into (3.81) we come to (3.71) with D . leads to the transformation of (3.57) to (3.71); therefore, the solution to (3.57) exists for all the time instances and is unique. To prove that the solution to (3.57) is ergodic, it is sufficient to verify that the following condition holds [71]: Z I D

1

1

1 exp  2 ./

²Z

 0

³ 2a. / d  d  < 1.  2 . /

(3.82)

Substituting the coefficients a and  from (3.78, 3.79) into (3.82) we get Z I D Z  

1 1

³ ² Z  fE ./ . / d  d  exp 2 2./ 0

1

1

1 c0

Z

0  1

1

7 3

R 0

fE ./ d  ® R ¯ d jF . /j exp 2 0 . / d 

fE ./ d  D

1 , c0

(3.83)

where c0 D 0  7If =3. Remark 3.2. By construction of the model (3.55)–(3.57) it is clear that it was actually supposed that the inertial subrange expands from 0 to 1. In realistic models, however, this range is finite. It is not difficult to improve the model (3.55)–(3.57) by writing it in a more general form: OL d O L , t /L , D V . dt

d 1 , D O L, t / dt T .

d L . / D a.L / d  C  .L / d W . /, where V ., t / and T ., t / are the characteristic velocity and the life time of the turbulent eddies of size , respectively. Note that the time t appears here since the flow, in general, is not stationary.

88

Chapter 3 Stochastic Lagrangian models and relative prodispersion

3.3.4 Example We present an example of the model (3.57) with fE ./ D p

1 exp¹ 2 =212 º. 212

In this case, from (3.67) we get Z F ./ D



1

fE . / d D  p exp¹ 2 =212 º. 2 1

For , we choose the function ./ D ,

1 <  < 1,

where  is a positive parameter, satisfying the condition < and

where 0 is determined from

1 , 212

(3.84)

7 0 > 0 1 , 6

(3.85)

1 1 D  . 2 20 212

(3.86)

Assumptions (a) and (b) are satisfied. Indeed, assumption (a) is straightforward, since (3.71) is in our case a Langevin equation which has a unique solution for all the time instances. To the assumption (b): ² ³ Z 1 1 0 t2 1 If D 0, (provided U.0/ < 0), we denote by C D C .l0 /,  D  .l0 / the roots of equations 2 3=2 F .l0 / C U.0/C C C D 0, 3

2 F .l0 / C U.0/  3=2 D 0, 3

(4.11)

respectively. By continuity of S t and (5), it follows from (4.9)–(4.10) that the random time  D  .l0 / satisfies l. .l0 // D 0,

P ¹ .l0 / <  .l0 / < C .l0 /º D 1,

(4.12)

provided U.0/ < 0. From (4.11), (4.12) it follows that R l0  .l0 / '

0

dl R1=2 .l/

j U.0/ j

R l0 '

0

. Ll /1=3 d l

.L"/ N 1=3

3 D 2 j U1 .0/ j j U1 .0/ j



l02 "N

1=3 ,

provided that l0  L, U1 .0/ < 0. Thus the random time  .l0 / behaves like  .l0 /  l2 . "N0 /1=3 as l0 ! 0 (provided that U1 .0/ < 0).

4.3 Mathematical formulation of a new model Again, let us consider a local isotropic high-Reynolds number turbulence. Inertial range (IR) is an important ingredient of such flows. Let l.t / be the instantaneous distance between two fixed fluid particles at time t . It is well known (see, e. g., [146]),

101

Section 4.3 Mathematical formulation of a new model

that essentially the eddies of size comparable to the separation of the particles are responsible for their relative dispersion. Let vl , l be a typical value of the velocity and the life-time of eddies of size l, respectively. Then in accordance with the above-stated argument we formulate the following hypothesis: Let  be a dimensionless time defined by the local relation d D

dt l.t/

under the initial condition  .0/ D 0. Then the dimensionless relative velocity of particles (more exactly, the longitudinal component of the relative velocity)  D

dl dt

vl.t/

is a universal stationary stochastic process (i. e., the statistical characteristics of this process do not depend on the turbulence parameters ", N L, TL , etc.). In IR, this hypothesis takes more precise form. Indeed, in accordance with the second similarity Kolmogorov’s hypothesis (see [146]) and using dimensional consider2 ations we have vl  .N"l/1=3 , l  . l"N /1=3 . Therefore, by the above-stated hypothesis in IR of the local isotropic high-Reynolds number turbulence the relative distance between two fluid particles, l.t /, satisfies the relation  1=3 d "N dl 1=3 , t > 0, (4.13) D .N"l/ ./, D 2 dt dt l with the initial conditions l.0/ D l0 ,  .0/ D 0; here l0 is the initial distance between the particles. In spite of some similarity between (4.3) and the first equation of (4.13) there is an essential difference between them. In (4.3), the instantaneous velocity ddtl depends only on l.t / while in accordance with (4.13) dl D .N"l.t //1=3  dt

Z t  0

"N l 2 .s/



1=3 ds

i. e., it depends on the whole trajectory ¹l.s/, 0  s  t º. In what follows we assume that ./ is a measurable, separable stochastic process almost certainly having continuous sample functions (for definitions see, e. g., [29]). Without loss of generality we also assume that "N D 1, l0 D 1 and consider the next dimensionless problem dX D X 1=3 ./, dt X.0/ D 1,

d D X 2=3 , dt

 .0/ D 0.

t >0

(4.14) (4.15)

102

Chapter 4 A new Lagrangian model of 2-particle relative turbulent dispersion

4.4 A qualitative analysis of the problem (4.14) for symmetric ./ 4.4.1 Analysis of the problem (4.14) in the deterministic case First, let us consider the problem dY dX D X 1=3 f .Y /, D X 2=3 , dt dt 0 < X , Y < 1, t > 0, X.0/ D 1,

Y .0/ D 0,

(4.16) (4.17) (4.18)

where f : .0, 1/ ! .1, 1/ is a continuous function. In accordance with the theory of ordinary differential equations, there exists a solution X D X.t /, Y D Y .t / to (4.16) which is defined in some time-interval Œ0, t / and such that the following alternative cases happen: 1. t D 1; 2. t < 1 and the solution X D X.t /, Y D Y .t / satisfies at least one of the conditions 2a. 2c.

lim X.t / D 0,

2b.

lim Y .t / D 0,

2d.

t!t t!t

lim X.t / D 1,

(4.19)

lim Y .t / D 1.

(4.20)

t!t t!t

This solution is called the maximal solution to (4.16), and the interval Œ0, t / is called the maximal interval. Lemma 4.1. There exists a unique maximal solution X D X.t /, Y D Y .t /, t 2 Œ0, t / to (4.16) with Z 1 Z 2 exp. S /d  , S D f .s/ds, (4.21) t D 3 0 0 and lim Y .t / D 1,

t!t

lim t!t X.t / D lim !1 exp.S /,

(4.22)

lim t!t X.t / D lim !1 exp.S /.

(4.23)

Proof. Let us suppose on the contrary, that there exist two maximal solutions Xi .t /, Yi .t /, t 2 Œ0, ti / .i D 1, 2/. It is obvious that the function Yi : Œ0, ti / ! Œ0, Yi / is monotonically increasing. Here, Yi D lim Yi .t /, t!ti

i D 1, 2.

Section 4.4 A qualitative analysis of the problem (4.14) for symmetric . /

103

Therefore, there exists a converse function Yi1 : Œ0, Yi / ! Œ0, ti / .i D 1, 2/. Let XN i .Y / D Xi .Yi1 .Y //,

YNi .Y / D Yi1 .Y /,

Y 2 Œ0, Yi / .i D 1, 2/.

Then, it is not difficult to show that XN i .Y /, YNi .Y / .i D 1, 2/ obey d YN d XN N D 1, YN .0/ D 0. (4.24) D XN f .Y /, D XN 2=3 , X.0/ dY dY Moreover, by definition, lim  XN i .Y / D lim Xi .t /, lim Yi .t / D lim Yi , lim  YNi .Y / D ti . Y !Yi

t!ti

t!ti

Y !Yi

t!ti

(4.25) Now, let us show that Yi D 1 .i D 1, 2/. Indeed, otherwise, if Yi < 1 then by (4.24), (4.25) ²Z Y ³ ² Z Y ³ Z Y i i 2  0 0 lim Xi .t / D exp f .Y /d Y , ti D exp f .y /dy d Y . 3 0 t!ti 0 0 (4.26) Since f ./ is continuous, (4.26) contradicts (4.19). Thus we have Yi D 1, i D 1, 2. Further, in view of the obvious uniqueness of the solution to (4.24), XN 1 .Y / D XN 2 .Y /,

YN1 .Y / D YN2 .Y /,

Y 2 Œ0, 1/.

(4.27)

Since ti D limY !1 YNi .Y /, i D 1, 2, then (22) yields t1 D t2 and Y1 .t / D Y2 .t /, t 2 Œ0, t /. Here t D t1 D t2 . Thus uniqueness of the maximal solution X D X.t /, Y D Y .t /, t 2 Œ0, t / to (4.16) is shown; (4.21) is an evident consequence of (4.26). Finally, by Yi D lim t!t Yi .t / D 1 and the first formula in (4.26) we get (4.22). The proof is completed.

4.4.2

Analysis of the problem (4.14) for stochastic ./

Now let us consider the problem (4.14) involving the stochastic function ./. The stochastic process ¹./,   0º is called symmetric if it has the same statistic ˇ characteristics as the process ¹./,   0º. For given real numbers ˛, ˇ let M˛ be a  –algebra generated by a family of random variables ¹./,T ˛    ˇº. A stochastic 1 D 1 process ¹./,   0º is called regular if the set M1 >0 M is trivial (i. e., 1 M1 consists of events having probability one or zero). It is well known that ¹./,   0º is regular if and only if sup j P .AB/  P .A/P .B/ j! 0,

A2M1

for every B 2 M01 .

 !1

104

Chapter 4 A new Lagrangian model of 2-particle relative turbulent dispersion

Theorem 4.1. Let ¹./,   0º be a symmetric regular stochastic process. Then with probability 1, the solution to .4.14/ is defined on the half-line RC D Œ0, 1/ and ® ¯ ® ¯ P lim X.t / D 0 D P lim X.t / D 1 D 0. (4.28) t!1

t!1

Proof. Let us define an event A, generated by the random variable ² ³ Z 1 2 exp S d  , t D 3 0 Z .s/ds S D 0 1 and P .A/ is equal to through A D ¹! : t .!/ < 1º. It is obvious that A 2 M1 0 or 1 due to regularity of ¹./,   0º. Let us show that P .A/ D 0. Indeed, let A0 D ¹! : t0 < 1º, where ² ³ Z 1 2 0 t D exp  S d  . 3 0

Then we have P .A/ D P .A0 / due to symmetry of ¹./,   0º. Further, by virtue of the inequality x C x1  2, for positive x we have 1

Z 0



2 S exp 3



 C exp

2  S 3

 d D 1

i. e., P .AA0 / D 0. Therefore P .A/ D P .A0 / D 0, i. e., P ¹t D 1º D 1. By Lemma 4.1 the last equality means that every solution of (4.14) is almost certainly defined on RC . To establish (4.28) let us consider two events: B D ¹ lim X.t / D 0º, t!1

and

C D ¹ lim X.t / D 1º. t!1

By Lemma 4.1 we have B D ¹ lim S D 1º, !1

C D ¹ lim S D 1º. !1

1 . By symmetry and regularity properties of ¹./,   0º It is obvious that B, C 2 M1 we have P .B/ D P .C / D 0 or 1. Since the events B and C are inconsistent, it is obvious that P .B/ D P .C / D 0. This completes the proof.

Let us give two examples which show that without the regularity or symmetry conditions, the assertion of Theorem 4.1 may be wrong.

Section 4.4 A qualitative analysis of the problem (4.14) for symmetric . /

105

Example 4.1. Let  be a symmetric real-valued random variable. Let ./  ,   0. Then it is obvious that ´ 1, 0 t D 2  3 ,  < 0. Therefore, 1 P ¹t < 1º D P ¹ < 0º D , 2 1 P ¹ lim X.t / D 0º D P ¹ < 0º D . t!t 2 In this example the regularity condition is broken. Example 4.2. Let

´  .1   /, . / D 0,

 2 Œ0, 1,  > 1.

– be a sequence of mutually independent random variables having the Let ¹ k º1 kD1 following distributions: P ¹ k D kº D

k2 , .k 2 C 1/

P ¹ k D k 3 º D

.k 2

1 , C 1/

k D 1, 2, : : : .

Let us define a stochastic process ¹./,   0º as ./ D . Œ C1  Œ  / .  Œ /,

  0,

 0. It is evident that the process ¹./,   0º where Œ  is the integral part of  , 0P is regular but not symmetric. From k P ¹ k D k 3 º < 1 we get by the Borel– Cantelly lemma that there exists an integer-valued random variable  such that P ¹ < 1º D 1, and (4.29)

k D k, k  . Since SkC D 

k 2 .3  2 / C . kC1  k / , 6 6

k D 1, 2, : : : ,

0  < 1,

(4.30)

then from (4.29) we get SkC D 

k 2 .3  2 / C , 6 6

From (4.31) it follows that ²Z P

1 0

²

k  ,

0  < 1.

³ ³ 2 exp S d  < 1 D 1, 3

(4.31)

106

Chapter 4 A new Lagrangian model of 2-particle relative turbulent dispersion

and P ¹ lim S D 1º D 1. !1

Consequently, by Lemma 4.1 P ¹t < 1º D 1, P ¹ lim X.t / D 0º D 1, i. e., the t!t assertions of Theorem 4.1 are broken. The following theorem makes the asymptotic behavior of X D X.t / as t ! 1 more precise. Theorem 4.2. Let ¹./,   0º be a symmetric regular stochastic process such that there exists a sequence ¹"n º1 nD1 of real-valued positive numbers and an increasing of real-valued positive numbers satisfying the following conditions: sequence ¹n º1 nD1 1 X

P ¹j SOn j< "n º < 1, and "n n ! 1, as n ! 1,

(4.32)

nD1

where n2 D M S 2n , SOn D

1 n S n ,

S D

R 0

.s/ds. Then

P ¹ lim X.t / D 0º D P ¹ lim X.t / D 1º D 1. t!1

t!1

(4.33)

Proof. Let A D ¹ lim X.t / D 0º,

B D ¹ lim X.t / D 1º,

A0 D ¹ lim S D 1º,

B 0 D ¹ lim S D 1º.

t!1

!1

t!1

!1

Then bySLemma 4.1 we have P .AA0 / D 0, P .BB 0 / D 0, where AA0 D .AnA0 / .A0 nA/. Therefore it is sufficient to establish that P .A0 / D P .B 0 / D 1. Indeed, by (4.32) and the Borel–Cantelly lemma there exists an integer-valued random variable , P ¹ < 1º D 1 such that j SOn j "n ,

n  ,

i. e., j S n j "n n ,

n  . P .A0

(4.34) B 0/

[ D 1. Because of Therefore, since "n n ! 1 as n ! 1, we have 0 the symmetry of ¹./,   0º, it is obvious that P .A / D P .B 0 /. Finally, from 1 and using the regularity of ¹./,   0º we get P .A0 / D P .B 0 / D 1. A0 , B 0 2 M1 This completes the proof. The following example shows that in the general case the conditions (4.32) can not be ignored.

Section 4.4 A qualitative analysis of the problem (4.14) for symmetric . /

107

Example 4.3. Let ¹./,   0º be a stochastic process defined as in Example 4.2, but with 1 P ¹ k D k 2 º D P ¹ k D k 2 º D 2 , 2k  1 1 P ¹ k D 1º D P ¹ k D 1º D 1 2 . 2 k Then by the Borel–Cantelly lemma there exists an integer-valued random variable , P ¹ < 1º D 1 such that j k jD 1, k  . Therefore from (4.30) and by symmetry of ¹./,   0º we have ³ ² ³ ² 1 1 D P lim S D  D 1, P lim S D !1 6 6 !1 and hence by Lemma 4.1 we get P ¹ lim X.t / D e 1=6 º D P ¹ lim X.t / D e 1=6 º D 1. t!1

t!1

Thus, the assertion (4.33) of Theorem 4.2 is not true. This is a consequence of violation of the condition (4.32). Now, to prove (4.4) we first obtain the following. Corollary 4.1. Let ¹.t /, t  0º be a UO process. Then .4.33/ is valid. Proof. Since h.t /.t C  /i D  2 exp.˛ /, the spectral density of .t /, p./, is equal to (see, e. g., [146]): 2 2 ˛ . p./ D .˛ 2 C 2 / It is known (see, e. g., [85]) that a stationary Gaussian process with the spectral density p./ is regular if and only if Z 1 j ln p./ j d  < 1. .1 C 2 / 0 It is obvious that for UO process .t / this condition is satisfied. Now, h.t /i D 0 implies that .t / is symmetric. To show (4.32), we choose n D n, "n D 1, n D 1, 2, : : : . Then  Z n 2 2 2 2 n D .t /dt D 2 .e ˛n  1 C ˛ n/, ˛ 0 i. e., n behaves like n  n1=2 as n ! 1. Therefore, the condition (4.32) is obviously true. Thus all the sufficient conditions of the Theorem 4.2 are true and therefore (4.33) implies the assertion of the corollary. The proof is completed.

108

Chapter 4 A new Lagrangian model of 2-particle relative turbulent dispersion

In the next section we generalize the results obtained for nonsymmetric processes ./.

4.5 Qualitative analysis of the problem (4.14) in the general case A stochastic process ¹./,   0º is said to be a process satisfying the strong mixed condition if ˛. / D sup

0

sup

A2M0 ,B2M1 C

jP .AB/  P .A/P .B/j ! !1 0.

When it satisfies the following stronger condition '. / D sup

0

sup

A2M0 ,P .A/¤0,B2M1 C

jP .B=A/  P .B/j ! !1 0,

then it is said that ¹./,   0º satisfies the uniformly strong mixed condition. Lemma 4.2. Let ¹./,   0º be a stochastic process satisfying the strong mixed be a monotone increasing sequence of real-valued positive condition; let ¹k º1 kD0 2kC1 1 numbers; let ¹Ak ºkD0 be a sequence of events such that Ak 2 M 2k , k D 0, 1, 2, : : : and 1 1 X X P .Ak / D 1, ˛.2k  2k1 / < 1. (4.35) kD0

kD1

Then P

\ 1 [ 1

Ak

D 1,

nD0 kDn

i. e., almost surely an infinite number of events of sequence Ak , k D 0, 1, 2, : : : happens. Proof. Let Ack D nAk . Then \ ² \ ³ ²[ ³ 1 1 1 [ 1 \ 1 Ak Ack D lim P Ack P n DP nD0 kDn

nD0 kDn

D lim

lim P

n!1

 nCm \

n!1 m!1

kDn

c Ak .

kDn

(4.36)

Section 4.5 Qualitative analysis of the problem (4.14) in the general case

109

From the strong mixed condition we get P

 nCm \

Ack



nCm Y

kDn

P .Ack / C

kDn

D

nCm X

˛.2k  2k1 /

kDnC1

nCm Y

nCm X

kDn

kDnC1

.1  P .Ak // C

˛.2k  2k1 /.

Consequently, lim P

 nCm \

m!1

Ack



kDn

1 Y

.1  P .Ak // C

kDn

1 X

˛.2k  2k1 /.

kDnC1

The properties (4.35) show that the rhs last inequality tends to 0 as n ! S1 T of the 1. Therefore by (4.36) we get P ¹ n. 1 nD0 kDn Ak /º D 0. This completes the proof. Lemma 4.3. Let ¹./,   0º – be a stochastic process satisfying the following conditions: 1. M . / D 0,   0, sup 0 M  2 . / < 1,  2 D M S 2 ! 1, as  ! 1, R where S D 0 .s/ds; 2. there exists a real-valued random variable , such that P ¹ < 0º > 0, P ¹ > 0º > 0; S P SO D ! , as  ! 1; (4.37)  3. there exist positive real numbers C , ı such that P¹

sup

m mC1

j ./ j< C º > ı,

m D 0, 1, 2, : : : ;

(4.38)

4. the strong mixed condition is valid. Then for every real number a > 0 ²Z P

1

³ exp¹aS ºd  D 1 D 1,

0

and P ¹ lim S D 1º D P ¹ lim S D C1º D 1. !1

!1

(4.39)

110

Chapter 4 A new Lagrangian model of 2-particle relative turbulent dispersion

Proof. Let S. , C  / D SC  S , conditions 1–2 of the theorem yield

 2 . , C  / D M S 2 . , C  /. Then the

P O ,  C / D S. ,  C / ! S.

, as  ! 1  . ,  C /

 2 . , C  / ! 1,

(4.40)

for every  0. Therefore by the conditions 3)–4) of the theorem there exists a monotone increasing sequence    < nk < nkC 14 < nkC 12 < nkC 34 < nkC1 <   

(4.41)

of positive integers such that for every k D 1, 2, : : :  2 .nk , nkC 14 /  .2n2k /4 , ˛.nkC 12

1  nkC 14 /  2 , k

n2kC 1  k 2 , 2

 2 .nkC 12 , nkC 43 /  .2n2kC 1 /4 , 2

1 n2kC1  .k C 1/2 , ˛.nkC1  nkC 34 /  2 , k 1 O º   > 0, P ¹S.nk , nkC 41 /  q  .nk , nkC 14 / O kC 1 , nkC 3 /  q P ¹S.n 2 4

1  .nkC 12 , nkC 43 /

º   > 0.

(4.42)

Here  is a positive real number (depending on the distribution of the random variable ). For the sake of simplicity we suppose that a  0. Let us define the following sequences of events: 1

Ak D ¹S.nk , nkC 14 /   2 .nk , nkC 14 /, 1

S.nkC 12 , nkC 34 /   2 .nkC 12 , nkC 43 /º, Ck D ¹j SnkC 1 j n2kC 1 º, 2

2

Bk D ¹j Snk j n2k º,

Dk D ¹

sup

nkC 1  nkC 1 C1 4

j ./ j< n2k º,

4

k D 1, 2, : : : . Then by the Chebyshev inequality we have P .Bk /  and consequently

X k

1 , k2

P .Ck / 

P .Bk / < 1,

1 , k2

X k

k D 1, 2, : : : ,

P .Ck / < 1.

(4.43)

Section 4.5 Qualitative analysis of the problem (4.14) in the general case

111

Due to (4.38) there exists a positive integer k0 such that P .Dk /  ı,

k  k0 .

(4.44)

By the strong mixed condition and by the last two inequalities of the set (4.42) we get 1

P .Ak /  P ¹S.nk , nkC 14 /   2 .nk , nkC 14 /ºP ¹S.nkC 12 , nkC 43 / 1

  2 .nkC 12 , nkC 43 /º  ˛.nkC 12  nkC 14 /   2  k D 1, 2, : : : .

1 , k2 (4.45)

By (4.43) and the Borel–Cantelly Lemma there exists an integer-valued random variable , P ¹ < 1º D 1, such that for every k   the following events Bkc D nBk and Ckc D nCk happen, i. e., j Snk j n2k ,

j SnkC 1 j< n2kC 1 , 2

2

k  .

(4.46)

Next from the strong mixed condition, the inequality ˛.nkC1  nkC 34 /  k12 , the set of inequalities (4.42) and by Lemma 4.2 it follows that almost surely there is an infinite , ¹Dk º1 . For the sake of simplicity number of events from all sequences ¹Ak º1 kD1 kD1 of notation, we suppose that almost surely there occur all events in the sequences , ¹Dk º1 . Then, denoting n0k D nkC 14 , k   we get ¹Ak º1 kD1 kD1 1

Sn0k D SnkC 1 D S.nk , nkC 14 / C Snk   2 .nk , nkC 14 /  n2k  n2k , 4

1

SnkC 3 D S.nkC 12 , nkC 34 / C SnkC 1   2 .nkC 12 , nkC 34 / C n2kC 1  n2kC 1 , 4 2 2 2 Z n0 C k .s/ds  n2k  0 sup 0 j .s/ j 0, Sn0k C D Sn0k C nk  nk C1

n0k

k  , 0   1. From the condition  2 ! 1 as  ! 1 and the last conditions we have the assertions of the lemma. The case a < 0 can be considered likewise. Hence, the lemma is proved. Here we make the following remark: Remark 4.1. For a stationary (in the narrow sense) process ¹./,   0º the condition (4.38) is always valid. This is a consequence of almost certain continuity of a sample function of ./. The direct consequence of the proved theorem is the following assertion.

112

Chapter 4 A new Lagrangian model of 2-particle relative turbulent dispersion

Theorem 4.3. Let ¹./,   0º be a stochastic process satisfying all the conditions of Lemma 4.3. Then almost certainly there exists a unique solution of the problem .4.14/ defined on RC and satisfying the condition .4.33/. In what follows we assume that M . / D 0,   0. Corollary 4.2. Let ¹./,   0º, be a stochastic process such that (i) it satisfies the uniformly strong mixed condition, (ii) the condition (4.38) is valid, and (iii) for some positive number ı > 0 sup M j ./ j2Cı < 1, 0

n2Cı n1 ! 1, as n ! 1.

Then almost certainly the solution to .4.14/ is unique, it is defined on RC and .4.33/ is valid. This assertion is a consequence of the Theorem 4.3 and the sufficient conditions of the central limit theorem (CLT) [8]. Using the sufficient conditions of the CLT [9] and by Theorem 4.3 we get the following assertion: Corollary 4.3. Let ¹./,   0º be a stationary (in the narrow sense) stochastic process such that (i) it obeys the uniformly strong mixed condition, (ii) there exists some ı > 0 such that M j .0/ j2Cı < 1, and (iii) n2 ! 1, n ! 1. Then all the assertions of Corollary 4.2 are valid. Thus we have established several sufficient conditions for ./ under which the stochastic model (4.14) has a unique solution defined for all positive time instances.

Chapter 5

The combined Eulerian–Lagrangian model

5.1

Introduction

The commonly used approach to the well-developed turbulence is the statistical description where the Eulerian velocity UE .x, t / is considered to be a 3D-random field. Having the samples of UE .x, t /, one determines the Lagrangian trajectory X.t , x0 /, t  0 of a fluid particle originating at position x0 at time t D 0 from the equation of motion: @X (5.1) D UE .X, t / @t subject to the initial condition X.t D 0, x0 / D x0 .

(5.2)

The Lagrangian velocity V.t , x0 / is related to the Eulerian velocity UE .x, t / by V.t , x0 / D UE .X.t , x0 /, t /.

(5.3)

Equation (5.1) can be integrated numerically in time; (5.3) means that the instantaneous particle velocity is the same as the fluid velocity at the instantaneous particle position. Each Eulerian quantity f .x, t / can be related to the Lagrangian quantity by F .x0 , t / D f .X.t , x0 /, t /,

x0 2 R3 ,

t  0.

A Lagrangian description allows us to analyze directly the motion of material fluid elements. Importance of the Lagrangian trajectories is that the quantities of practical interest are expressed through the n-particle statistical characteristics [152, 157]. In particular, the mean concentration of a passive scalar and its covariance are defined through the 1-particle and 2-particle statistical characteristics, respectively [146,214]: Z hc.x, t /i D p1L .x, t ; x0 /S.x0 /d x0 , Z hc.x, t /c.x0 , t /i D p2L .x, x0 , t ; x0 , x00 /S.x0 /S.x00 /d x0 d x00 , where c.x, t / is the concentration of the passive scalar released from an instantaneous deterministic source S.x0 /. The functions p1L and p2L are the 1-point and

114

Chapter 5 The combined Eulerian–Lagrangian model

2-point p. d. f.’s: p1L .x, t ; x0 / D hı.x  X.t , x0 /i, p2L .x, x0 , t ; x0 , x00 / D hı.x  X.t , x0 //ı.x0  X.t , x00 //i. Here and throughout all the chapter the symbol hi means the average over the realizations of the Eulerian velocity field. We are dealing here with homogeneous stationary incompressible high-Reynolds number turbulence. According to the Kolmogorov’s similarity hypotheses [146], in this case there exists a parameter "N (called mean rate of dissipation of kinetic energy) such that the statistical characteristics of the increments of the Eulerian velocity after normalyzing by a quantity v become universal functions (for all kind of fully developed turbulence) of a dimensionalless time t = and dimensionalless spatial distance r= provided r  L, t  T . Here L and T are the external spatial and temporal scales, respectively; is the Kolmogorov’s spatial microscale which is related to the velocity microscale v and the time microscale  by N 1=4 ,  D =v ,

D . 3 =N"/1=4 , v D . "/ where  is the kinematic viscosity of the flow. Another important property of the velocity field is that the 1-point distributions are approximately Gaussian while the structure of the 2-point distributions is more complicated. In particular, the p. d. f. of the velocity increments at two points separated by r is essentially non-Gaussian if r  . The same is true in the lower part of the inertial subrange [160]. Unfortunately, there are no adequate models which satisfactorily govern the turbulent velocity field described. Therefore, different approximations are constructed. For instance, in [102, 113, 198], spectral models of Gaussian random velocity fields were used to numerically solve the problem .5.1/. In [160, 170, 262], the Eulerian velocity field was obtained by the numerical solution of the Navier–Stokes equation. All these Eulerian approximations have drawbacks: the models [102, 113, 198] are pure Gaussian, while the models [160, 170, 262] are restricted to moderate values of the Reynolds number. In addition, the latter models need a great deal of computer time. Thus after choosing an appropriate approximation to UE .x, t /, and numerically constructing the set of Lagrangian trajectories, one calculates the desired Lagrangian statistical characteristics by averaging. This method we call for short the E-scheme. There is another method for simulation of Lagrangian trajectories which we call the Lagrangian scheme (for short, L-scheme). This approach is based on approximaO , x0 /. The relevant approxtions of the stochastic Lagrangian velocity V.t , x0 /, say V.t imate Lagrangian trajectory is then obtained from the ordinary stochastic differential equation O dX O , x0 /, (5.4) D V.t dt

115

Section 5.1 Introduction

O 0 , 0/ D x0 . Note that this is generally a nonlinear equation, since the random with X.x O /. Most frequently models O , x0 / itself depends on the random position X.t process V.t of the type .5.4/ are written in the form of stochastic differential equations (SDE) of the Ito type. These equations are much easier to solve numerically than the general equations of fluid particle motion .5.1/ in random fields. O , x0 / D V.t , x0 / The models .5.1/ and .5.4/ define one and the same flow if V.t for all Lagrangian trajectories. It would be an ideal approximation if we could find V.t , x0 /. Of course, this is impossible to do even for the simplest flows. However, when constructing models of the type .5.4/, it is reasonable to use some relations between the statistics of the Eulerian velocity field UE .X, t / and of the 6-dimensional Lagrangian field Y.t , x0 / D .X.t , x0 , V.t , x0 //. For instance, in the case of incompressible flows there is an integral relation between the probability density functions of UE .x, t / and Y.t , x0 / (see [153]; a more general situation of compressible flows is treated in [154]). The E-scheme is rigorous; in this scheme the particles are moving in a self-consistent way. In particular, the following relation holds: Z (5.5) P1L .x, vE, t ; x0 / D P2L .x, vE, x0 , vE0 , t ; x0 , x00 /d x0 d vE0 , where P1L , P2L are the 1- and 2-particle Lagrangian p. d. f.’s of the E-scheme: P1L .x, vE, t ; x0 / D hı.x  X.t , x0 //ı.E v  V.t , x0 //i, and P2L .x, vE, x0 , vE0 , t ; x0 , x00 / D hı.x  X.t , x0 //ı.E v  V.t , x0 //ı.x0  X.t , x00 //ı.E v 0  V.t , x00 //i. However in the L-scheme, this relation is generally not satisfied. Therefore, in [238] it was noted that when constructing 2-particle Lagrangian models by the L-scheme, it is desired to satisfy the relation .5.5/. This principle is called in [238] a “two-to-one” reduction. As far as we know, there is no L-scheme which satisfies this principle. The incompressibility of UE leads to certain restrictions on the L-scheme. In particular, the following integral relation was derived by Novikov [153]: v1 , : : : , vEn ; x1 , : : : , xn / PnE .E Z D PnL .x1 , vE1 , : : : , xn , vEn , t ; x01 , : : : , x0n /d x01 : : : d x0n , where PnE , PnL are the n-point Eulerian and the n-particle Lagrangian p. d. f.’s, respectively: v1 , : : : , vEn ; x1 , : : : , xn / D PnE .E

n DY iD1

E ı.E vi  UE .xi , t // ,

116

Chapter 5 The combined Eulerian–Lagrangian model

PnL .x1 , vE1 , : : : , xn , vEn , t ; x01 , : : : , x0n / n n E DY Y ı.xi  X.t , x0i // ı.E vj  V.t , x0j // . D iD1

j D1

In [169] this restriction is introduced as a consistency condition. In the cases n D 1, 2, this restriction was used by Thomson [237,238] (called by him a well-mixed condition) when constructing 1- and 2-particle L-schemes. Note also that Novikov [155] used his integral relation for n D 2 to construct stochastic Lagrangian models of the relative dispersion of two particles. Remark 5.1. It should be noted that the practical interest of the L-scheme comes from the fact that it provides the desired behavior of the statistics of relative dispersion in the inertial subrange. In the E-scheme, this a one of the main problems. Let us give a brief overview of the work on the 2-particle models. There are many 1-dimensional 2-particle models (e. g., see [44, 45, 123, 129, 213–218, 237], etc.). As mentioned in [114,238], these models do not adequately describe the 3D turbulent dispersion. As far as 2-particle 3D models in high-Reynolds-number isotropic stationary turbulence is concerned, we mention the Eulerian model presented in [62, 113, 198], and two Lagrangian models: Thomson’s model [238], the Borgas-Sawford model [15] and the combined Eulerian–Lagrangian model proposed in [91]. Let us consider some features of these models. In the Eulerian model [62, 113] n-particle Lagrangian characteristics can be calculated in a self-consistent way in the sense that a “m to k” reduction principle holds which means that the k-particle p. d. f.’s (for k < m) are the marginal p. d. f.’s of the m-particle p. d. f.’s. However, its Gaussian nature leads to the failure of the relative dispersion statistics in the inertial subrange. In the models [15,238] this is not the case, but another difficult problem arises: there is no unique choice of the coefficients specifying the model, and the “two-to-one” reduction principle does not hold. Models which take into account the statistical structure of the velocity in the inertial subrange were developed in [44,45,108,155,236]. Note that the 1-dimensional models have no adequate physical interpretation. However quasi-1-dimensional models which describe the evolution of the distance between two fluid particles in 3D space have clear physical meaning. We call a model of relative dispersion quasi-1-dimensional if it describes the evolution of the distance .t / between two fluid particles in the space R3 [114]. Since the quasi-1-dimensional model describes the motion of two particles in 3D, it is reasonable to require that this model is consistent with the integral relation (5.6) in a sense given in the next subsection. As to the Eulerian model, the Kaplan–Dinnar’s model [91] also deals with n particles and satisfies the “m to k” reduction principle. However, the well-mixed condition for the 2-particle model is not satisfied (see the Appendix to this chapter).

117

Section 5.2 2-particle models

In this chapter, we construct a new combined Eulerian-Lagrangian model which is believed to be free of the drawbacks of all the models mentioned above. It is based on a modification of our model (see [109, 113]) and a Lagrangian 3D model of relative dispersion [110].

5.2

2-particle models

5.2.1

Eulerian stochastic models of high-Reynolds-number pseudoturbulence

The numerical models of high-Reynolds-number pseudoturbulence were used in [102]. In [144], a randomized model was constructed for a simplified case, and in [153, 191], a randomized model of 3-dimensional incompressible Gaussian stationary velocity with the Kolmogorov spectral density frozen in time was created. In this section we modify this model by introducing correlation in time. The correlation is chosen in accordance with the Kolmogorov similarity hypotheses. Let us assume that the Eulerian velocity field UE .x, t / has the following partial energy spectrum:  t , (5.6) E.k, t / D E.k/'  .k/ where E.k/ D E.k, 0/ is the 3-dimensional energy spectrum; ',  are given functions such that '.0/ D 1, and '.t / is an even positive definite function of t , i. e., Z 1 exp.i !t /'.t /dt  0. '.!/ Q D 1

In addition it is assumed that  .k/ > 0. These assumptions come from the condition that  E.k, t / kj kl  ‰j l .k, t / D ı , jl 4k 2 k2

j , l D 1, 2, 3,

(5.7)

is a partial spatial-temporal spectral tensor of an isotropic random field. The energy spectrum is defined by ´ C1 "N2=3 k 5=3 , kmin  k  kmax , (5.8) E.k/ D 0, otherwise with the normalization

Z

1

3 E.k/d k D u20 . 2 0 For the frequency part, we use four models. Model 1 is  ² ³ t 1 1 2 2 2  .k/ D , ' D exp  k0 u0 t . k0 u0  .k/ 2

(5.9)

(5.10)

118

Chapter 5 The combined Eulerian–Lagrangian model

In the next three models we choose  .k/ D .N"k 2 /1=3 ,

(5.11)

so that in Model 2  '

t  .k/



² ³ t 1=3 2=3 D exp  ˛ D e ˛"N k t ,  .k/

(5.12)

and in Model 3 ²  2 ³  t  t 2=3 4=3 2 ' D exp  ˛ D e ˛"N k t .  .k/  .k/

(5.13)

Finally, Model 4 is: 

t '  .k/



²

³  t t 1=3 2=3 D exp ˛ cos  D e ˛"N k t cos."N1=3 k 2=3 t / . (5.14)  .k/  .k/

In formulas .5.12/–.5.14/, ˛  0,   0 are some parameters of the model. The quantity "N is the rate of energy dissipation, kmin and kmax are the minimal and maximal wave numbers, respectively. Thus kmin D 2=L where L is the external length scale, C1 is a universal constant (C1 D 1.4). It is easy to verify that for all four models the relevant correlation tensor is positive definite. The input parameters of the models are: "N, L, kmax , u0 , and k0 (Model 1), plus the quantity ˛ in Models 2, 3 and plus  in Model 4. These parameters are related by the condition .5.9/, so we consider "N to be a dependent parameter. To construct the simulation formula for the velocity field, it is necessary to have the spatial-temporal spectra ˆlm .k, !/ (see Section 1.2.4 in [191]). From 1 ˆlm .k, !/ D 2

Z

1 1

‰lm .k, t /e i!t dt .

(5.15)

From .5.6/, .5.7/ we find for the model p D 1, 2, 3, 4: ˆlm .k, !/ D Flm .k/qp .k, !/, where

k D jkj D

 E.k/ kl km Flm .k/ D ılm  2 , 4k 2 k

p

k12 C k22 C k32 ,

l, m D 1, 2, 3,

(5.16)

119

Section 5.2 2-particle models

and q1 .k, !/ D p

² ³ 1 !2 exp  2 2 , 2k0 u0 2k0 u0

˛ "N1=3 k 2=3 , .˛ 2 "2=3 k 4=3 C ! 2 / ² ³ 1 !2 exp  , q3 .k, !/ D 1=3 2=3 p " k ˛ 4˛"2=3 k 4=3 q2 .k, !/ D

q4 .k, !/ D

˛ "N1=3 k 2=3 .˛ 2 "N2=3 k 4=3 C 2 "N2=3 k 4=3 C ! 2 / Œ˛ 2 "N2=3 k 4=3 C ."N1=3 k 2=3  !/2 

Œ˛ 2 "N2=3 k 4=3

1 . C ."N1=3 k 2=3 C !/2 

Now we can write the randomized model of the homogeneous stationary isotropic Gaussian random field UE .X , t / specified by the spatial-temporal spectral tensor p ˆlm .k, !/ given by .5.16/ (for the derivation of the simulation formula, see [191]): UE .x, t / D

n p X

° ± p p En .j j / cos. j / C .j j / sin. j / ,

(5.17)

j D1 p

p

.1/

.2/

.3/

where j D kj .j , x/ C !j t , and j D . j , j , j /, j D 1, : : : , n are independent 3-dimensional isotropic unit vectors for models p D 1, 2, 3, 4; j D .1/

.2/

.3/

.1/

.2/

.3/

.j , j , j / and j D . j , j , j / are mutually independent standard Gaussian vectors; kj , j D 1, : : : n are random variables with the densities 8 1 < E.k/, k 2 j , pj .k/ D En : 0, otherwise, Z

with En D

E.k/d k,

j

j D 1, : : : , n,

and j , j D 1, : : : , n are nonoverlapping segments which compose a partition of  D .0, 1/, the support of the spectrum. For each fixed k D kj , j D 1, : : : , n, the p random quantities !j , p D 1, : : : , 4, j D 1, : : : , n are distributed with the densities qp .kj , !/, j D 1, : : : , n. The partition of the spectrum support D

n [ j D1

j ,

j

\

l D ;,

j ¤ l,

120

Chapter 5 The combined Eulerian–Lagrangian model

where j D ŒkQj , kQj C1 /, j D 1, : : : , n, k1 D kmin , kQnC1 D kmax , is chosen so that (e. g., see [191]) Z Z 1 E.k/d k D E.k/d k. n

j From this, we find  h j  j 2=3 i3=2 2=3 kQj C1 D kmin 1  , C kmax n n

j D 1, : : : , n.

The random numbers kj are simulated by h 2=3  kj D kQj

i3=2 u20  , j ,1 nC1 "N2=3

j D 1, : : : , n,

(5.18)

p

and the random frequencies !j are simulated as p !j1 D k0 u0 2 ln j ,2 sin.2j ,3 /,   2=3 !j2 D ˛ "N1=3 kj tan j ,4   , 2 q p 2=3 !j3 D 2˛ "N1=3 kj 2 ln.j ,5 sin.2j ,6 /, and

8    ˇL), we come to the system .5.19/. Thus, ˛ and ˇ are free parameters of the model.

5.3.2

Models for the p. d. f. of the Eulerian relative velocity .rel/

Thus to specify our model, we need the p. d. f. pE . Unfortunately, the information .rel/

about the Eulerian p. d. f. pE is usually approximate, taken from measurements or calculations. Therefore, we have to use some analytical formula approximating the Eulerian p. d. f. For simplicity, we construct the p. d. f. fE which is defined in .5.25/ so k that .5.26/ is satisfied. The function fE is chosen in a form of a sum of three Gaussian densities   ² .1  q0 / .1 C x/ 1 2 .  a1 /2 k p exp  C q0 exp  fE ./ D p 2C 2 2 2 2 2 C  2 ³ .  a2 / .1  x/ 1 p exp  C , (5.30) 2 2 2 2

124

Chapter 5 The combined Eulerian–Lagrangian model

where C is the universal constant in Kolmogorov’s law of two thirds, and ° 2.1  x/2 ±1=3 .1 C x/ a1 D , a2 D  a1 , 5q0 x.1 C x/ .1  x/ ° .1 C x/ 2 .1  x/ 2 ±1=2 . a1  a2 D C 2 2

(5.31)

The parameters are chosen so that 0  q0 < 1, and x 2 .x0 , 1/, where x0 D x0 .q0 / D 2 3 1=2 . This choice ensures that the moments satisfy the known relations .1 C 25 4 q0 C / [114, 146] Z 1 Z 1 k k fE ./ d  D 1, fE ./ d  D 0, 1

Z

1

1

Z

k

 2 fE ./ d  D C ,

1 1

4 k  3 fE ./ d  D  . 5 1

Note that the free parameters q0 and x can be chosen to satisfy some additional constrains for the fourth and fifth cumulants. Indeed, let us consider the excess E and a quantity R defined by Z 1 Z 1 k k  4 fE ./d  D C 2 .E C 3/,  5 fE ./d  D C 5=2 .R C 10S /, 1

1

where S D 4=.5C 3=2 / is the asymmetry. From .5.30/–.5.31/ we find  2 4=3 1 2.3x 2  1/ , ED 1=3 2 1=3 4=3 5 C 2 q0 .1  x / x  2 5=3 1 4.x 2 C 1/ RD . 2=3 2 2=3 5 C 5=2 q0 .x.1  x // Thus if the quantities E and R are given, the values of x and q0 can be found from the latter relations. We assume that the transverse component of the relative velocity is Gaussian. From 2 i D 2C 0 .N"r/2=3 where C 0 D 4C =3. The function the Kolmogorov two thirds law, hv? ? fE takes in this case the form ² ³ 1 2 exp  fE ./ D . 2 C 0 2C 0 .rel/

Remark 5.2. In the case of Gaussian pE plicitly. Indeed, in this case we have ² ³ 2 1 k exp  fE ./ D p , 2C 2 C

, the coefficient a.r, vE/ can be found ex-

fE? ./ D

² ³ 1 2 exp  . 2 C 0 2C 0

125

Section 5.4 Appendix

From .5.27/, .5.29/, and .5.23/ we find  2 1=3 ²

  ³ "N v2 C0 C 0 vk 4 vk 7 r vE a.r, vE/ D   C C . r 3 4C .N"r/1=3 .N"r/2=3 r 3 .N"r/1=3 C 0 .N"r/1=3

5.4

Appendix

The 2-particle model [91] is given in a finite-difference form: q 2 .t / .x.k/ .t //, vE.k/ .t C t / D RL .t /E v .k/ .t / C 1  RL x.k/ .t C t / D x.k/ .t / C vE.k/ .t /t ,

k D 1, 2,

with the initial conditions vE.k/ .0/ D .x.k/ .0//, k D 1, 2. Its continuous analog can be written as d x.k/ D vE.k/ .t /, dt d vE.k/ vE.k/ C D dt TL

(A.1) r

2 .x.k/ .t /, t /, TL

k D 1, 2.

Here  is a Gaussian random field with h i .x, t / j .x0 , t 0 /i D Cij .r/ı.t  t 0 /, i , j D 1, 2, 3, ° ri rj ± Cij .r/ D  2 .r/ıij C .r/ 2 , r D x  x 0 , r where  and  are functions of the distance r D jx0  xj:  D g.r/,

1  D f .r/  g.r/, g.r/ D f .r/ C rf 0 .r/, 2  1=3 2 r . f .r/ D 1  2 r C L2

Let us introduce the column vectors: X D .x.1/ , x.2/ /T ,

V D .E v .1/ , vE.2/ /T .

Then the pair of systems .16.47/ is written as dX D V, dt V dV C ‚.X, t /, D dt TL

(A.2)

126

Chapter 5 The combined Eulerian–Lagrangian model

where

r ‚.X, t / D

! 2 .x.1/ , t / . TL .x.2/ , t /

Hence, 2 h‚.X, t /‚ .X , t /i D TL T

0

0

9 8 .1/ 0.1/ .1/ 0.2/ ˆ > ˆC.x  x / C.x  x /> > ˆ > ˆ ;, : C.x.2/  x0.1/ / C.x.2/  x0.2/ /

where C is the matrix with the entries Cij , i , j D 1, 2, 3. We noe need to derive the Fokker–Planck equation. We present here the general case given in [96]. Let us consider the system of random equations dxi D fi .x, t / C i .x, t /, dt

.i D 1, : : : , n/,

(A.3)

where fi are deterministic functions and i are Gaussian random fields such that hi .x, t /j .x0 , t 0 /i D Kij .x, x0 , t /ı.t  t 0 / .i , j D 1, : : : , n/. Let p.x, t ; x0 / D hı.x  x.t ; x0 //i be the transition density, where x.t ; x0 / is the solution of the system of random equations .A.3/ with the initial conditions x.0; x0 / D x0 . Then the Fokker–Planck equation for the density p has the form ¯ 1 @2 ® @ ŒAi .x, t / p  @p Bij .x, t /p , D C @t @xi 2 @xi @xj where

ˇ 1 Kij .x, x0 , t / ˇˇ Ai .x, t / D fi .x, t / C ˇ0 , 2 @xj x Dx

and Bij .x, t / D Kij .x, x, t /. Thus in our case the Fokker–Planck equation reads 1 @ 2 @pL @2 @pL  .V˛ pL / D .†˛ˇ pL /, C V˛ @t @X˛ TL @V˛ TL @V˛ @Vˇ where k†˛ˇ k6˛,ˇ D1

8 9 I R> ˆ ˆ > D: ;, RI

RD

1 C 2

and I is a 3 3-identity matrix. As commonly used, we consider the case ² ³ p 1 3  exp  2 ˛ˇ V˛ Vˇ , pE .V / D .2 / 2

(A.4)

127

Section 5.4 Appendix

where ˛ˇ are the entries of the matrix ƒD

k˛ˇ k6˛,ˇ D1

1 8 I R91  ˆ > 6 > D k†˛ˇ k˛,ˇ D1 Dˆ : ; , RI

and  D det ƒ. Let us introduce the notation  D 1  2 ,

D 1  . C /2 .

Then through some algebra we find D 9 8 A B > ˆ >, ˆ ƒD: ; BA

1 ,

 2

A D .I  R2 /1 ,

B D RA,

where A D k˛ij k3i,j D1 ;

B D kˇij k3i,j D1 ,

with 1 2 C  2 ri rj , ˛ij D ıij C   r2  . C / C  ri rj . ˇij D  ıij    r2 To derive the well-mixed condition, we need the following representations which can be found by differentiating the function pE : ² ³ 1 1 @ˇ  @ ln  @pE D pE  2 Vˇ V , @X˛ 2 @X˛  @X˛ 1 @pE D  2 pE ˛ˇ Vˇ , @V˛  ² ³ 2 1 1 @ pE D  2 pE  ˛ˇ C 2 ˛ V ˇ ı Vı . @V˛ @Vˇ   The well-mixed condition requires that pE satisfies .16.48/. Since pE does not depend on t , this implies that

 1 @ˇ V˛ 1 @ ln  6 1  2 Vˇ V  pE  pE ˛ˇ Vˇ V˛ pE 2 @X˛  @X˛ TL TL  2 ² ³ pE †˛ˇ 1 D  ˛ˇ C 2 ˛ V ˇ ı Vı . (A.5) TL 

128

Chapter 5 The combined Eulerian–Lagrangian model

When considering the terms with equal powers of the velocity we note that the “zeropower terms” are equal: 6pE =TL D pE †˛ˇ ˛ˇ =TL . However the linear and cubic terms can not vanish, hence, the well-mixed condition is not satisfied for this model.

Chapter 6

Stochastic Lagrangian models for 2-particle relative dispersion in high-Reynolds-number turbulence A new stochastic model (a diffusion approximation) for the relative dispersion of pair of particles in high-Reynolds-number incompressible turbulence is proposed. An attempt is made to uniquely define the coefficients of SDE governing the relative dispersion process under a closure assumption about the quasi-1-dimensional model. An approach for constructing a diffusion approximation of the relative dispersion taking into account the intermittency is proposed.

6.1

Introduction

To model processes such as mixing and combustion we need statistical information about the relative displacement of particle pairs in a turbulent flow (e. g., [44,214]). In this chapter we will consider the process of relative dispersion of two particles (briefly, the relative dispersion process) in small spatial and temporal regions compared to the external scale of turbulence. For the high-Reynolds number flows, in these scales the Kolmogorov similarity hypotheses hold [146]. Therefore, universal laws are true for the relative dispersion process. Classical results due to Batchelor in the problem under discussion are well known [10]. However a series of unknown functions and constants in this theory is yet not explicitly known. Therefore, there is a much interest in constructing models which enable to find the desired statistical characteristics of the relative dispersion process. There are several stochastic Lagrangian 2-particle models of turbulent dispersion. survey of literature on the given subject can be found in [114, 146]. In [237], a new approach to construct stochastic Lagrangian models which are consistent with the incompressibility was proposed. This approach deals with a generalized Langevin equation governing the dynamics of the particle’s motion whose coefficients satisfy the so-called well mixed condition. It should be noted that this condition does not define uniquely the stochastic model even in the case of isotropic stationary turbulence [15]. Therefore, it is important to find out physically plausible constraints on these coefficients. In this chapter we propose an assumption which, together with the well-mixed condition due to Thomson, lead to a uniquely defined stochastic model of relative dispersion in the isotropic or locally isotropic turbulence. This approach treats the case of high-Reynolds-number turbulence with intermittency in a general manner.

130

Chapter 6 Stochastic Lagrangian models for 2-particle relative dispersion

6.2 Preliminaries Throughout this chapter we follow the statistical approach to turbulent flows, i. e., we regard the flow as a member of an ensemble of flows with identical external conditions (e. g., [146]) and consider only ensemble average quantities. Such averages will be denoted by angled brackets. Thus the velocity field UE .x, t / is considered as a sample from 3D random field. Let UE .x, t / be a velocity field in a high-Reynolds-number turbulent flow. As commonly used, we assume that the relative dispersion process .r.t /, vE.t // can be approximated by a 6D Markov diffusion process .Or.t /, vEO .t // governed by the SDE p (6.1) d rO D vEO dt , d vEO D a.t , rO , vEO / dt C 2C0 "N d W.t /, where a D .a1 , a2 , a3 / is a 3D vector function to be defined; C0 is a universal constant in the linear law for the structure function of the Lagrangian velocity [146], "N is the mean rate of dissipation of turbulence kinetic energy, W.t / is the standard 3D Wiener N 1=2 is commonly used (e. g., see process. The choice of the diffusion coefficient .C0 "/ 15, 237, 238). We briefly present the Thomson’s approach for the determinig the drift coefficient a in .6.1/. In [153], the following relation was derived: Z v ; r, t / D pL .r, vE; t , r0 / d r0 , (6.2) pE .E where v ; r, t / D hı.E v  UE .x C r, t / C UE .x, t //i pE .E is the probability density function (p. d. f.) of the relative Eulerian velocity UE .x C r, t /  UE .x, t / and v  vE.t ; r0 //i pL .r, vE; t , r0 / D hı.r  r.t ; r0 //ı.E is the joint p. d. f. for the separation r.t ; r0 / and relative velocity vE.t ; r0 /. Here ı./ is the Dirac delta function, r.t / D r.t ; r0 / is the separation vector between two particles which initially were separated by r0 , and vE.t / D vE.t ; r0 / is the relative velocity. We denote by .Or.t ; r0 /, vEO .t ; r0 // the solutions to .6.1/ under the initial conditions rO .t D t0 / D r0 ,

vEO .t D t0 / D vE0 ,

(6.3)

v  vE0 /i D pE .E v ; r0 , t0 /. Then where vE0 is the 3D random variable with the density hı.E the p. d. f. v  vEO .t ; r0 //i pOL .r, vE; t , r0 / D hı.r  rO .t ; r0 //ı.E solves the Kolmogorov–Fokker–Planck equation @ @2 pOL @pOL @pOL C v i i C i .ai pOL / D C0 "N i i . @t @r @v @v @v

Section 6.3 A closure of the quasi-1-dimensional model of relative dispersion

131

Here and in what follows we use the summation convention. Since the true p. d. f. pL satisfies Novikov’s relation .6.2/, it is natural to require that the p. d. f. of our model, pOL , also satisfies this R relation. From this, integrating the last equality with respect to r0 , and changing pOL d r0 with pE we come to the well-mixed condition by Thomson [153]: @ @2 pE @pE @pE (6.4) C v i i C i .ai pE / D C0 "N i i . @t @r @v @v @v This equation can be considered as a restriction on a, pE fixed. It is clear that .6.4/ does not define a uniquely. As mentioned in [15], the problem of the unique choice of a is nontrivial. For, even in the case of stationary isotropic turbulence it is possible to find at least two models of the type .6.1/ with different coefficients a satisfying the wellmixed condition .6.4/ which lead to essentially different behavior of the mean squared separation between two particles in the inertial subrange. Therefore it is of interest to formulate new constraints on a originating from physically plausible arguments. Such an attempt will be made in the next section.

6.3

A closure of the quasi-1-dimensional model of relative dispersion

In [114], a quasi-1-dimensional process of relative dispersion (for short: Q1D-process) is defined as a stochastic process simulating the distance r.t / between two particles. A diffusion model for Q1D-process in [45] was proposed for analysis of the temperature fluctuations in decaying isotropic turbulence. Further study of this process was undertaken in [108,109,114]. It turns out that for Q1D models a Q1D analog of the 3D well-mixed condition is true [114]. This enables us to construct a diffusion approximation of the Q1D process in the form O vO k /d W .t /, d rO D vO k dt , d vO k D X.t , rO , vO k / dt C Y .t , r,

(6.5)

where X./, Y ./ are some coefficients, W .t / is a 1-dimensional standard Wiener process (see [109, 114]). Let us consider the Q1D-process r.t O / D .Or.t /, rO .t //1=2 determined by the 3D model .6.1/. Then the 2D process .Or.t /, vO k .t // satisfies the SDE d rO D vO k dt , where

d vO k D .aO k C

2 p vO ? N Wk .t /, /dt C 2C0 "d r

(6.6)

rO .t / 2 D .vEO , vEO /  vO k2 , /, vO ? aO k D ak .t , rO , vEO / D .a.t , rO , vEO /, r.t O /

and Wk .t / is the 1D Wiener process which is related to W.t / through d Wk .t / D rO .t/ .d W.t /, r.t/ /. O

132

Chapter 6 Stochastic Lagrangian models for 2-particle relative dispersion

Note that in general case the system .6.6/ is not closed with respect to the variables r, O vO k . For instance, in [15, 238] the models are not closed in this sense. In this chapter we make the following assumption about the model .6.1/: Assumption 6.1. The Q1D model .6.6/ is closed relative to the variables r, O vO k , in the sense that v2 (6.7) ak .t , r, vE/ C ? D X.t , r, vk / r for some function X depending only on the shown arguments. Remark 6.1. Note that there is a full analogy between this assumption and the hypothesis of a closed description of the 1-particle vertical dispersion in vertically inhomogeneous surface layer of the atmosphere (e. g., see [237]). In the next section we show that in the case of isotropic turbulence Assumption 6.1 together with the well-mixed condition .6.4/, lead to a unique choice of the coefficient a in .6.1/.

6.4 Choice of the model .6.1/ for isotropic turbulence Thus we assume that the Eulerian velocity field UE .x, t / is an incompressible isotropic v ; r, t / D pE .vk , v? ; r, t /, and the coefficient a.t , r, vE/ random field. In this case, pE .E has the general form r a.t , r, vE/ D '.vk , v? ; r, t / C r

vE .vk , v? ; r, t / , v

(6.8)

where

q r v , vE/1=2 , vk D .E v , /, v? D v 2  vk2 . r D .r, r/1=2 , v D .E r Under Assumption 6.1 we have 'C

v2 vk C ? D X.t , r, vk /. v r

(6.9)

Then the well-mixed condition .6.4/ takes the form  2 v k v? 2vk 1 @ @ @pE @pE 2 .XpE / C vk C C pE C  pE v @t @vk @r r v? @v? ? v r ² 2  ³ @ pE 1 @ @pE D C0 "N C v . (6.10) ? @vk2 v? @v? @v?

133

Section 6.4 Choice of the model .6.1/ for isotropic turbulence

Integrating this equality over v? with a weight 2v? and taking into account that X does not depend on v? , we find k

k

k

@p @pE @2 pE 2vk k @ k .XpE / C vk E C , C pE D C0 "N @t @vk @r r @vk2 Z

where k pE .vk ; r, t /

(6.11)

1

D 0

2v? pE .vk , v? ; r, t / dv? k

is the p. d. f. of the longitudinal Eulerian relative velocity uE .r, t / D .UE .x C r, t /  UE .x, t /, rr /: k

k

pE .vk ; r, t / D hı.vk  uE .r, t //i. Here we assumed that  2 v?

ˇ 2 v k v? @pE ˇˇ N ? D 0.  pE  C0 "v v r @v? ˇjv? j!1

Under the additional assumption ˇ  2 v? vk k ˇˇ D0 'C C pE ˇ v r vk !1 we obtain X.t , r, vk / D C0 "N

F .vk ; r, t / @ k ln pE .vk ; r, t /  k @vk pE .vk ; r, t /

(6.12)

(6.13)

(6.14)

by integrating .6.11/ over vk . Here, Z F .vk ; r, t / D

vk

1

²

³ k k 2vk0 k @p @pE C vk0 E C pE .vk0 ; r, t / dvk0 . @t @r r

(6.15)

Thus the coefficient X./ in .6.6/ is determined. We find the coefficients ' and from .6.9/ and .6.10/. Integrating .6.10/ over v? with the weight v? we get vk v v @ ln pE .vk , v? ; r, t / C .vk , v? ; r, t / D C0 "N v? @v? r 

v G.vk , v? ; r, t / , 2 v? pE .vk , v? ; r, t /

(6.16)

where G.vk , v? ; r, t / ³ Z v? ² 2vk @ @pE @pE 0 0 0 .XpE / C vk dv? . (6.17) D C C pE .vk , v? ; r, t / v? @t @vk @r r 0

134

Chapter 6 Stochastic Lagrangian models for 2-particle relative dispersion

From .6.9/, .6.14/, and .6.16/ we get '.vk , v? ; r, t / D C0 "N

vk @ @ k ln pE .vk ; r, t /  C0 "N ln pE .vk , v? ; r, t / @vk v? @v? F .vk ; r, t /



k

pE .vk ; r, t /

C

vk G.vk , v? ; r, t / v2  . 2 v? pE .vk , v? ; r, t / r

(6.18)

Thus Assumption 6.1 leads to the unique choice of the coefficient a through the Eulerian p. d. f. pE under quite general conditions .6.12/–.6.13/. Let us consider the case when k

? .v? ; r, t /. pE .vk , v? ; r, t / D pE .vk ; r, t /pE

(6.19)

Under this assumption the formulas .6.16/–.6.18/ can be simplified. Indeed, from .6.11/ we find k (6.20) G.vk , v? ; r, t / D pE .vk ; r, t /F1 .vk , v? ; r, t /, where Z F1 .vk , v? ; r, t / D

v? 0

²

³ ? ? @pE @pE 0 0 0 0 dv? . .v? ; r, t / C vk .v ; r, t / v? @t @r ?

(6.21)

Substituting the value G from .6.20/ in .6.18/ we get by .6.19/ '.vk , v? ; r, t / D C0 "N

F .vk ; r, t / @ k ln pE .vk ; r, t /  k @vk pE .vk ; r, t /

 C0 "N

(6.22)

vk @ vk F1 .vk , v? ; r, t / v 2 ?  . ln pE .v? ; r, t / C 2 ? v? @v? r v? pE .v? ; r, t /

Analogously, from .6.20/, .6.16/, and .6.19/ it follows vk v v @ ? ln pE .v? ; r, t / C v? @v? r vF1 .vk , v? ; r, t / .  2 ? v? pE .v? ; r, t /

.vk , v? ; r, t / D C0 "N

(6.23)

Thus we have finished the construction of the model .6.1/ for the isotropic turbulence.

135

Section 6.5 The model of relative dispersion of two particles

6.5

6.5.1

The model of relative dispersion of two particles in a locally isotropic turbulence Specification of the model

In this section we deal with incompressible flow with a large Reynolds number. Let us consider the relative dispersion process in the scales from the inertial subrange which are much less than the external turbulence scale. In this case one uses an approximation called local isotropic which means that the distributions of the velocity increments are isotropic and stationary [146]. Since in the inertial subrange the unique external parameter is "N, we have the following representation for the Eulerian p. d. f. pE :  vk 1 v? v ; r, t / D pE .E v ; r/ D fE , pE .E , (6.24) "r N .N"r/1=3 .N"r/1=3 where fE .k , ? / is a universal dimensionless function of dimensionless arguments k 2 .1, 1/, and ?  0. Due to the local isotropy, we seek the coefficient a in .6.1/ in the form  a D a.r, vE/ D

"N2 r

1=3 ²  'Q

 ³ vk v? v? r Q vE , , C , (6.25) 1=3 1=3 1=3 1=3 r v .N"r/ .N"r/ .N"r/ .N"r/ vk

where '. Q k , ? /, Q .k , ? / are universal dimensionless functions. We find these functions assuming the function fE to be given. We restrict our considerations to the case .6.19/. The general case can be also treated analogously. The condition .6.19/ can be written, due to .6.24/, in the form k

fE .k , ? / D fE .k /fE? .? /,

(6.26)

k

where fE .k /, fE? .? / are dimensionless universal functions normalized so that Z 1 Z 1 k fE ./ d  D 1, 2fE? ./ d  D 1. 1

0

k

? in .6.19/ satisfy the relations The dimensional functions pE , pE   vk 1 1 v? k k ? ? f f pE .vk ; r/ D , pE .v? ; r/ D . (6.27) .N"r/1=3 E .N"r/1=3 .N"r/2=3 E .N"r/1=3 2 =r in .6.6/. By To find the coefficient a we first derive the coefficient X D ak C v? the definition of F (see .6.15/) and from the relations .6.27/ it is easy to get ³ ² 2 Z vk .N"r/1=3 7 k k k k fE ./ d  , k D . (6.28)  fE .k / C F .vk ; r, t / D r 3 3 1 .N"r/1=3

136

Chapter 6 Stochastic Lagrangian models for 2-particle relative dispersion

Consequently, by .6.14/,  vk "N2=3 Q X , r 1=3 .N"r/1=3 R k k 7 k k2 d ln fE 3 1 fE ./ d  Q C  . X.k / D C0 k d k 3 f .k / X.vk ; r, t / D

(6.29)

E

When dealing with the quasi-1-dimensional model .6.6/ it is convenient to use the O 1=3 . Then from .6.29/ and .6.6/ we obtain dimensionless velocity Ok D vO k =.N"r/ d rO D .N"r/ O 1=3 Ok dt , d Ok D .

p "N 1=3 O "N / a0 .k /dt C . 2 /1=6 2C0 d W .t /, 2 rO rO

where Q k/  a0 .k / D X.

k

k2

d ln fE  D C0 3 d k

To determine the coefficients ' and the relation .6.27/ it follows that

7 3

(6.30)

R k

k 1 fE ./ d  k fE .k /

.

(6.31)

, we first find F1 . By the definition .6.21/ and

F1 .vk , v? ; r, t / D 

2 vk v? p ? .v? ; r/. 3r E

Substituting this expression for F1 and F given by .6.28/ into .6.22/–.6.23/ we find  '.vk , v? ; r/ D  .vk , v? ; r/ D

"N2 r "N2 r

1=3  'Q 1=3

v? , , .N"r/1=3 .N"r/1=3  vk v? Q , , .N"r/1=3 .N"r/1=3 vk

(6.32)

where k

d ln fE  '. Q k , ? / D C 0 d k

7 3

R k

and

k 1 fE ./ d  k fE .k /

q

 C0

2 k2 C ? d ln f ?

k d ln fE? 2  .k2 C ? /, (6.33) ? d ?

4k q 2 2 . k C ? (6.34) ? d ? 3 In the case of Gaussian p. d. f. pE these formulas have a very simple form. The condition .6.19/ is automatically satisfied in this case. Note that the integral in .6.33/ can be explicitly found. After some algebra one can see that a.r, vE/ is quadratically dependent on vE. Q .k , ? / D C0

E

C

137

Section 6.5 The model of relative dispersion of two particles

6.5.2

Numerical analysis of the Q1D-model .6.30/

In this section we compare the model .6.30/ against the models [15, 238] and analyse how the non-Gaussianity of pE affects the constant g in the cubic Richardson law. k As fE , we choose a sum of three Gaussian densities, specified by two parameters, q0 and x:   ² .1  q0 / 2 .  a1 /2 .1 C x/ 1 k fE ./ D p p exp  C q0 exp  2C 2 2 2 2 2 C  2 ³ .  a2 / .1  x/ 1 p exp  C , (6.35) 2 2 2 2 where C is a universal constant in the Kolmogorov law of two-thirds and ² ³ 2.1  x/2 1=3 .1 C x/ , a2 D  a1 , a1 D 5q0 x.1 C x/ .1  x/ ² ³ .1 C x/ 2 .1  x/ 2 1=2 . a1  a2 D C 2 2

(6.36)

The parameters q0 and x are chosen so that 0  q0  1, and x 2 .x0 , 1/, where 2 3 1=2 . The choice of quantities .6.36/ ensures that x0 D x0 .q0 / D .1 C 25 4 q0 C / Z 1 Z 1 k k fE ./ d  D 1, fE ./ d  D 0, 1 1 Z 1 Z 1 4 k k  2 fE ./ d  D C ,  3 fE ./ d  D  , (6.37) 5 1 1 k

provided q0 > 0. The reason of this choice is that the true fE satisfies .6.37/ (see [114, 146]). Note that choosing appropriately the two free parameters q0 and x, it is possible k to control the excess and the fifth moment of fE . In our numerical calculations, we studied the behavior of the constant g in the Richardson law    N 3 1Co , hr 2 .t0 C  /i D g " 0 k

2=3

 0 D

r0 , "N1=3

(6.38)

for different models of fE and the values of C0 . This dependence for the Gaussian pE is well studied in [15]. In Figure 6.1 we show the dependence g D g.C0 / for the Gaussian pE (q0 D 0), and compare it against the relevant results obtained in [15] for the Thomson and Borgas–Sawford models. In the range C0  27, the difference between our Gaussian model and the latter models appear to be more than 10 times. Thus Assumption 6.1 results in models which qualitatively differ from the models [15, 238].

138

Chapter 6 Stochastic Lagrangian models for 2-particle relative dispersion g

4 4

10

2



4 b C

1

4 2 4

b C

b C

Thomson’s model Borgas-Sawford’s model diffusion limit independet case model .6.30/

4 b C 2 4

b C

2 4

4 2 b 4 C b C 2b C 2b C

0.1 1

10 C0

Figure 6.1. Inertial subrange dispersion constant g as a function of C0 for the model .6.30/,bn.6.35/ with q0 D 0, for Thomson’s model (see [15], the model (4.3)), for Borgas– Sawford’s model (see [15], the model (7.6) with ' D 0.4), for the diffusion limit (see [15]), and for the independent-particle case.

In Figure 6.2, the dependence of g on C0 is shown for the Gaussian case (i. e., q0 D 0.), and for five non-Gaussian models .6.35/–.6.36/, for five values of the parameter pairs .q0 , x.q0 //: .1.0, 0.929/, .0.8, 0.917/, .0.6, 0.9/, .0.4, 0.874/, .0.2, 0.827/. This choice ensures that all the five models have the same excess E D 0.5. The data in Figure 6.2 show that for C0  3 the dependence g D g.C0 / violates the kinematic constrain tg.C0 /  4C0 . Therefore, it seems that our model is applicable for C0  3. Note that the Gaussian and non-Gaussian cases do not qualitatively differ. In the more accepted range of C0  2–7, the function g D g.C0 , q0 / depends on C0 (q0 fixed) much more strongly than on q0 , while C0 is fixed. Therefore, since we have no precise information about the values of C0 , there is no evidence to prefer nonGaussian pE .

Section 6.6 Model of the relative dispersion in intermittent locally isotropic turbulence 139 g 30 b C

25

q0 q0 q0 q0 q0 q0

2 4 ?

20 15

b C 2 4 ?

10 5

1

2

3

b C 2 ? 4

4

b C 2 4 ? 5

D 0.0 D 0.2 D 0.4 D 0.6 D 0.8 D 1.0

b C 2 4 ?

b C 2 4 ?

b C 2 ? 4

b C 2 ? 4

C ?b 4 2

6

7

8

9

C ?b 2 4 10

C0 Figure 6.2. Inertial subrange dispersion constant g as a function of C0 for the model .6.30/, .6.35/, for different values of q0 .

6.6

Model of the relative dispersion in intermittent locally isotropic turbulence

In this section we attempt to take account of intermittency. In the intermittent turbulence, the relation .6.24/ for the Eulerian p. d. f. pE takes the form (e. g., see [3, 146]) v ; r/ D pE .E

 vk 1 v? r , , fE , "r N .N"r/1=3 .N"r/1=3 L

(6.39)

where fE .k , ? , / is a dimensionless function of dimensionless arguments k 2 .1, 1/, ?  0, > 0. As in the case considered above, we assume that the process of relative dispersion .r.t /, vE.t // can be approximated by a Markov diffusion process .6.1/. By Assumption 6.1 and the well-mixed condition .6.10/ we obtain  a.r, vE/ D

"N2 r

1=3 ²  'Q

 ³ vk v? r r v? r vE Q , , , , C , .N"r/1=3 .N"r/1=3 L r .N"r/1=3 .N"r/1=3 L v (6.40) vk

140

Chapter 6 Stochastic Lagrangian models for 2-particle relative dispersion

where 'Q and Q are some dimensionless functions uniquely defined by fE . For simplicity, we consider the case k

fE .k , ? , / D fE .k , /fE? .? , /.

(6.41)

Then by .6.22/,.6.23/ we find k 7 Ik .k , / @ ln fE k @ ln fE? 2  3 k  .k2 C ? /  C0 @k  @ ? ? fE .k /  2 1 @Ik k 1 @I?  k , (6.42) C 2 ? ? fE @ f @

'. Q k , ? , / D C0

Q .k , ? , / D C0

q 2 k2 C ? @ ln f ? E



? @? q 2 2 k k C ? @I

?

2f? ? E

@

E

C

4k q 2 2 k C ? 3

,

(6.43)

where Z Ik .k , / D

k

1

k fE ., / d ,

Z I? .? , / D

?

0

fE? ., / d .

For the coefficient X.vk , r/ D X.vk , r, "N, L/ in the model .6.6/ we have X.vk ; r/ D where XQ .k , / D C0

vk "N2=3 Q r , /, X. 1=3 1=3 L r .N"r/

k 7 k2 Ik @ ln fE 1 @Ik C  3 k  k @k 3 fE fE @

(6.44)

.

Remark 6.2. Note that for the intermittent turbulence, in addition to to ", N the parameter L enters the model. Hence, this model seems to give results which should differ from the classical Batchelor’s theory of relative dispersion (see [10, 146]). In particular, when considering the normalized longitudinal relative velocity 1 .vk .t0 C  , r0 /  vk .t0 , r0 // v k D .N" /1=2 r2

k

0 the p. d. f. hı.k  v /i of this velocity, due to the Batchelor thethen, for  "N1=3 ory, depends on k and does not depend on  . In our model .6.1/, .6.40/ the p. d. f. depends both on k and =T , where T D L2=3 =N"1=3 . Note that the model described in Section 6.5 fits Batchelor’ theory.

Section 6.7 Conclusions

6.7

141

Conclusions

A new stochastic model (a diffusion approximation) for the relative dispersion of a pair of particles in high-Reynolds-number incompressible turbulence was proposed. An attempt was made to uniquely define the coefficients of SDE governing the relative dispersion process under a closure assumption about the quasi-1-dimensional model. The coefficients of the diffusion approximation .6.1/ are uniquely expressed through the Eulerian p. d. f. pE . The relative dispersion in the inertial subrange of the locally isotropic turbulence was treated in detail. Using the stationarity and the second Kolmogorov similarity hypothesis, simple expressions for the coefficients of the diffusion model were obtained. In this case, a numerical study of the influence of the non-Gaussinity of pE on the statistical characteristics of the model, namely, on the constant g D g.C0 / in the Richardson cubic law .6.38/ was carried out. Qualitatively, the Gaussian and non-Gaussian models do not lead to a remarkable difference; the variation of the constant g when varying different models of pE , C0 fixed, is much less than its change when varying C0 in an acceptable range 2  C0  7. We conclude that there is no evidence for preferring non-Gaussian models since there is no exact information about the true value of C0 . In addition, an approach to constructing a diffusion approximation of the relative dispersion taking into account the intermittency ws proposed. This approximation requires further investigation and comparison with the new results in the problem under study (e. g., see [155, 156]).

Chapter 7

Stochastic Lagrangian models for 2-particle motion in turbulent flows. Numerical results It is shown that the relative diffusion in a stationary incompressible Gaussian isotropic random field does not exhibit the Richardson cubic law. A 2-particle combined Eulerian–Lagrangian stochastic model which correctly reflects the behavior in the inertial subrange is developed. In addition, in this model, the “two-to-one” reduction due to Thomson is satisfied with high accuracy except for a small initial time interval where the error is slightly higher.

7.1 Introduction In this chapter we use the notion of pseudoturbulence which is understood as a stationary incompressible Gaussian isotropic random field with a given spatial-temporal spectral tensor [146]. Throughout the chapter we assume that the mean velocity is zero. For brevity, we will call the pseudoturbulence with the Kolmogorov–Obukhov energy spectrum in the inertial subrange a classical pseudoturbulence. It is well known that the random velocity fields are often used to model different types of turbulent flows. For instance, S. Orszag has used a numerical scheme based on cubic formulas of the relevant spectral representation in the form of integrals with respect to the Gaussian measure. In [102], a randomized model of incompressible Gaussian isotropic random velocity field with a given energy spectrum was constructed for simulating Lagrangian trajectories and calculating some Eulerian and Lagrangian statistical characteristics. This approach was then used and developed by many authors (e. g., see [62, 63, 113, 128, 139, 191, 198]). In all these papers, spectral representations of the Gaussian random field were used. We mention also a wavelet expansion of the pseudoturbulence which was tried in [49]. It should be noted that it was generally believed that when choosing the Kolmogorov–Obukhov energy spectrum in the inertial subrange, this model gives a correct behavior of the 1- and many-particle Lagrangian statistical characteristics. For instance, it was assumed that the pseudoturbulence model will properly simulate the relative diffusion of two particles in the inertial subrange, and in particular, the Richardson cubic law N 3, (7.1) hr 2 .t /i ' g "t is true, provided that

 hr 2 .t /i1=2  L,

and

r0  hr 2 .t /i1=2 .

143

Section 7.2 Classical pseudoturbulence model

Here r.t / is the distance between two particles, r0 is the initial distance, "N is the mean rate of dissipation of kinetic energy, is the internal and L is the external characteristic spatial scales of turbulence; g is a universal constant. Attempts to confirm numerically the Richardson law were made in [63,198]. In these papers, the numerical results have not shown clear evidence of the reproduction of the universal Richardson law. In [51] it was reported that in 2D pseudoturbulence, the cubic law expands over even many decades of scaling behavior. Our calculations, however, have shown that for the classical pseudoturbulence the Richardson law does not hold. It turns out that the mean square separation can be approximated by the power law N 3 Œk0 ."t 3 /1=2  , hr 2 .t /i ' g "t

(7.2)

where k0 D 2=L,  and g are dimensionless constants which generally depend on the time correlation structure. This confirms our assumption that the Gaussian nature of the pseudoturbulence leads to the failure of the relative dispersion statistics in the inertial subrange. In turn, this leads to incorrect microscale behavior of the concentration fluctuations. Note that the study of the strong influence of the small scale relative dispersion on the intensity fluctuations is of current interest [44,214]. To construct a 2particle stochastic model which correctly reflects the behavior in the inertial subrange we made an attempt [199], suggesting a combined Eulerian–Lagrangian model. In this chapter we present the calculation details, which confirm that the classical pseudoturbulence model does not mimick the Richardson law (at least for the model Reynolds numbers about 109 ), while the combined model reproduces the Richardson law with high accuracy. In addition, the new combined model manifests another important property due to Thomson [238], the so-called “two-to-one” reduction principle. We applied both stochastic models to the calculation of the intensity of concentration fluctuations for an instantaneous 3D Gaussian source, which gives considerably different results for times larger than the characteristic external time scale.

7.2

Classical pseudoturbulence model

7.2.1

Randomized model of classical pseudoturbulence

Let us assume that the Eulerian pseudoturbulent velocity field UE .x, t / has the following partial spatial-temporal spectral tensor (e. g., see [146]): ‰j l .k, t / D

E.k/  kj kl  a"N1=3 k 2=3 t ı e  , jl 4k 2 k2

j , l D 1, 2, 3,

(7.3)

where E.k/ is the energy spectrum, a is a dimensionless constant which controls the time decorrelation, and k D jkj.

144

Chapter 7 Stochastic Lagrangian models for 2-particle motion in turbulent flows

The energy spectrum is defined by ´ C1 "N2=3 k 5=3 , E.k/ D 0, Z

with the normalization

k0  k  kmax , otherwise,

(7.4)

1

3 E.k/d k D u20 , (7.5) 2 0 where C1 ' 1.4 is the universal constant in the Kolmogorov–Obukhov five-thirds law and 3u20 D hjUE j2 i is the energy of turbulence. In the model, the following input parameters are involved: "N is the mean rate of dissipation of kinetic energy, k0 , kmax are the minimal and maximal wave numbers, respectively, and a is a dimensionless parameter characterizing the velocity decorrelation in time. The inner and external spatial scales of our model are D 2=kmax , and L D 2=k0 , respectively. Therefore, since the Reynolds number is expressed  4=3 , it is naturally in our case to define the model by (e. g., see [61, 146]) Re  L  kmax 4=3 O D . Reynolds number as Re k0 In [199], we presented four variants of the spatial-temporal spectral tensor with different decorrelation functions. The first variant is that used by Kraichnan [102], and in the three other cases the decorrelation time scale was dependent on the wave number. All these three cases have not shown a qualitative difference; therefore in the present chapter we have chosen the partial spatial-temporal spectral tensor in the form (7.3). The general simulation formula of the pseudoturbulent velocity field with the tensor (7.3) is [191, 199]: UE .x, t / D

n X p

° ± Ej .j j / cos. j / C .j j / sin. j / ,

(7.6)

j D1 .1/

.2/

.3/

where j D kj .j , x/ C !j t , and j D . j , j , j /, j D 1, : : : , n are in.1/ .2/ .3/ dependent 3-dimensional random isotropic unit vectors ; j D .j , j , j / and .1/

.2/

.3/

j D . j , j , j / are mutually independent standard Gaussian random vectors; kj , j D 1, : : : n are random variables with the densities 8 < 1 E.k/, k 2 j , Ej pj .k/ D : 0, otherwise, Z

with Ej D

E.k/d k,

j

j D 1, : : : , n

and j , j D 1, : : : , n are nonoverlapping intervals which compose a partition of  D .k0 , kmax /, the support of the spectrum. For each fixed k D kj , j D 1, : : : , n,

145

Section 7.2 Classical pseudoturbulence model

the random quantities !j , j D 1, : : : , n are distributed with the densities q.kj , !/, j D 1, : : : , n where q.k, !/ D

a"N1=3 k 2=3 , .a2 "2=3 k 4=3 C ! 2 /

! 2 .1, 1/.

In [199], we have chosen the partition of the spectrum support n [

D

j ,

j

\

l D ;, j ¤ l,

j D1

where j D ŒkQj , kQj C1 /, j D 1, : : : , n, kQ1 D k0 , kQnC1 D kmax , so that Z

j

1 E.k/d k D n

Z E.k/d k.

(7.7)



From this,  h j  j 2=3 i3=2 2=3 kQj C1 D k0 1 , C kmax n n

j D 1, : : : , n.

The random numbers kj are simulated by h 2=3 kj D kQj 

i3=2 u20  , j ,1 nC1 "N2=3

j D 1, : : : , n,

(7.8)

and the random frequencies !j are simulated as 2=3

!j D a"N1=3 kj

®  ¯ tan  j ,2  0.5 .

(7.9)

In the above formulas, j ,l , (j D 1, : : : , n, l D 1, 2) are mutually independent random numbers uniformly distributed in Œ0, 1. Modified algorithm In this chapter we use a different version of the simulation formula. First, we divide the interval Œk0 , kmax / into n0 parts ŒkOi , kOiC1 /, .i D 0, : : : , n0  1/ uniformly in logarithmic scale: kOi D k0 Qi , i D 0, : : : , n0 , where Q is chosen so that kOn0 D kmax . Then in each subinterval ŒkOi , kOiC1 / we apply the same subdivision (energy uniformly) as in the formula .7.7/. This algorithm provides better statistics in all parts of the energy spectrum. The number of simulated harmonics in this model equals n0 n and is proO In the next section we present the calculations of the mean square portional to ln.Re/. 2 separation hr .t /i.

146

Chapter 7 Stochastic Lagrangian models for 2-particle motion in turbulent flows

7.2.2 Mean square separation of two particles in classical pseudoturbulence Having the samples of UE .x, t / constructed by the modified algorithm, we determine a pair of Lagrangian trajectories Xi .t / D X.t ; x0i , t0 /, t 2 .1, 1/ of fluid particles originating at positions x0i at time t D t0 , i D 1, 2, from the equation of motion: @Xi D UE .Xi , t / @t

(7.10)

Xi .t0 / D x0i .

(7.11)

subject to the initial condition Let r.t / D jX2 .t /  X1 .t /j be the distance between two particles initially separated N a, k0 , kmax , the turbulence characterby r0 D r.t0 /. Generally, hr 2 .t /i depends on ", istics, and on t , r0 . Keeping in mind the Richardson cubic law, it is natural to expect that if the condition

D

2 2  hr 2 .t /i1=2  L D , kmax k0

and

r0  hr 2 .t /i1=2 ,

(7.12)

is satisfied, the mean square separation hr 2 .t /i will depend only on t , "N, and the dimensionless parameter a. In this case, by dimensional analysis we come to the Richardson cubic law .7.1/ with the constant g D g.a/. Hence, by fixing a, we can try to check whether or not the Richardson law holds for classical pseudoturbulence. In what follows we use the CGS units, and fix "N D 1 cm2 =s3 . Then, for simplicity, we omit the dimension units (e. g., we simply write "N D 1). In Figure 7.1 we present .t / D .hr 2 .t /i/1=2 in the log-log scales as a function of time, for three different values of r0 : 104 , 103 , and 102 (curves 1, 2, and 3) with the value of kmax equal to 105 , 104 , and 103 , respectively. The following parameters are here fixed: a D 1, k0 D 0.1. The value of n0 was chosen equal to the number of decades in the interval Œk0 , kmax /: n0 D 6, n0 D 5, and n0 D 4 for the curves 1, 2, and 3, respectively. The number of harmonics in each subinterval is n D 25. The number of simulated pairs of trajectories is: N D 13,000, N D 38,000 and N D 10,000 for the curves 1, 2, and 3, respectively. To make easy the comparison with the cubic behavior, we also show the straight line 4 which represents the cubic law. As is clearly seen from Figure 7.1, all three curves tend to the power law 2 D at ˇ with a D 0.0127 and ˇ D 3.75 (see the straight line 5, which represents the power law 2 / t 3.75 ). Note that the smaller the initial distance is, the broader the time interval where the power law (the corresponding part of the curve is parallel to the straight line 5) approximates the relevant curve (e. g., for curve 1 this is the interval .0.1, 10/). If we tried to find an interval for each curve where 2 .t / behaves cubically, it would imply that the cubic law depends on the initial distance, which is not true.

147

Section 7.2 Classical pseudoturbulence model 100

10 5 1

.t /

4

0.1

0.01

0.001

3 2 1

0.0001 0.01

0.1

1 Time t

10

100

Figure 7.1. The function .t / D hr 2 .t /i1=2 for different initial separations: r0 : 104 , 103 , and 102 (curves 1, 2, and 3) with the value of kmax equal to 105 , 104 , and 103 , respectively, and k0 D 0.1, a D 1. The cubic law 2 / t 3 (curve 4) and the power law 2 / t 3.75 (curve 5).

Note that in [198] we reported calculations showing that it is likely that the cubic law exists; but the initial distance was relatively large (r0 ' 0.01) (compare with curves 3 and 4 in Figure 7.1). In [198], we had no chance to carry out calculations for smaller values of r0 , since the direct randomization method (without the modification described above) would require some thousands of harmonics to provide stable calculations. To illustrate the accuracy achieved by our simulation technique, we could also plot the relative statistical errors (measured via the standard Monte Carlo 3 law) whose values are too small to be clearly seen in the picture: they vary along time from less than 5 % in curve 2 to a maximum of 10 % in the curves 1 and 3. Note that generally, the dependence N 3 f .k0 .N"t 3 /1=2 / hr 2 .t /i ' g "t is true, provided that hr 2 .t /i r02 ,

hr 2 .t /i 2 .

(7.13)

From Figure 7.1, we can estimate that f ./ /   with  ' 0.5 (for 0.03    3), which yields .7.2/.

148

Chapter 7 Stochastic Lagrangian models for 2-particle motion in turbulent flows 100 10 1 1 .t /

0.1 2

0.01 0.001 0.0001 0.01

0.1

1 Time t

10

100

Figure 7.2. The function .t / D hr 2 .t /i1=2 for k0 D 0.1 (curve 1) and k0 D 0.01 (curve 2); r0 D 104 , kmax D 105 , a D 1.

Concerning the dependence of .t / on k0 plotted in Figure 7.2 for k0 D 0.1 (curve 1) and k0 D 0.01 (curve 2), the results show that 2 ' 0.013t 3.75 and 2 ' 0.003t 3.75 , respectively. For both curves, r0 D 104 and a D 1, the number of trajectories N D 15,000. From this we conclude that the dependence .7.2/ holds under the conditions .7.12/. It is interesting to study whether or not the power dependence .7.2/ holds when varying the value of a. Remarkably, the change of a in .0, 1/ does not affect the behavior of hr 2 .t /i. For larger values of a the situation is different. In Figure 7.3 we present .t / for a D 10. Here, by 1 we indicate the curve for r0 D 104 , k0 D 0.1, kmax D 105 , and by 2 we denote the curve with r0 D 0.01, k0 D 104 , kmax D 103 . The number of trajectories N D 15,000. From these curves we can conclude that the power low .7.2/ is approximately true with  D 0.16. Note that from these results the dependence on k0 in .7.2/ is also confirmed (with accuracy to within 10 %). The number of harmonics in Figures 7.2 and 7.3 was taken as the number of decades in the interval .k0 , kmax / multiplied by 25. Thus we conclude that the classical pseudoturbulence model gives a stable power law in the inertial subrange which is different from the classical Richardson cubic law. It should be stressed that we are dealing here with high model Reynolds numbers in the range of 106 + 109 .

Section 7.3 Calculations by the combined Eulerian–Lagrangian stochastic model

149

1,000 100 10 .t /

1 0.1 0.01

0.001 0.0001 0.01

2 1 0.1

1

10

100

Time t Figure 7.3. The function .t / D hr 2 .t /i1=2 for a D 10 with k0 D 0.1, kmax D 105 , r0 D 104 (curve 1); and k0 D 104 , kmax D 103 and r0 D 0.01 (curve 2).

7.3

7.3.1

Calculations by the combined Eulerian–Lagrangian stochastic model Mean square separation of two particles

In this section we present numerical results for the mean square separation of two particles using the combined Eulerian–Lagrangian stochastic model. We show that the Richardson law is well satisfied in this model. We use the following notation: R.t / D ŒX1 .t / C X2 .t /=2 is the center of the two particle coordinates, and r.t / D X2 .t /  X1 .t / is the separation vector. The combined Eulerian–Lagrangian 2-particle model [199] has the form ²  ³  1 r r dR D UE R C , t C UE R  , t . dt 2 2 2   r °   r ± r  r ± ° dr D UE R C , t  UE R  , t C 1  vE.t /, (7.14) dt L 2 2 L p d vE.t / D a.r, vE/dt C 2C0 "N d W.t /, where UE is the pseudoturbulent velocity taken in the form .7.6/ with the energy uniform subdivision .7.7/, and a.r, vE/ is defined by [199]: ² ³ v 2 i r h 4 vk C0 i vE C 0 vk "N2=3 h 7   C C a.r, vE/ D 1=3 . 3 4C .N"r/1=3 .N"r/2=3 r 3 .N"r/1=3 C 0 .N"r/1=3 r   v j, Here vk D vE, rr is the longitudinal component of the relative velocity vector, v D jE 0 C ' 2 is the universal constant in the Kolmogorov two-thirds law, C D 4C =3, and

150

Chapter 7 Stochastic Lagrangian models for 2-particle motion in turbulent flows

C0 is the universal constant in the linear law for the structure function of the Lagrangian velocity [146]. The function is defined through 8 ˆ 0 < y < ˛, ˇ, where 0 < ˛ < ˇ < 1, and  is an arbitrary monotonically increasing function so that is continuous on .0, 1/. We recall that in .7.14/, W.t / is a standard 3D Wiener process. Thus in this model, two additional parameters are involved: the constant C0 and the function . We choose as a piecewise linear function defined by the constants ˛ and ˇ. These adjusting parameters were chosen as ˛ D 0.05 and ˇ D 0.1. Some remarks concerning this choice will be given in the next section. The exact value of the constant C0 is not known, it belongs to the interval (2, 8) (e. g., see [170, 262]). Note also that results [261] obtained by DNS suggest that C0 lies between 4 and 8. In Figure 7.4 we show the function .t / D hr 2 .t /i1=2 in log-log coordinates obtained for C0 D 4 (left panel) and C0 D 6 (right panel). The parameters were chosen as follows: k0 D 0.01, kmax D 1,000, the number of harmonics in the spectral representation .7.6/ was taken to be equal to 10. The number of samples was N D 4,000. In Figure 7.4 the upper curve 1 corresponds to the initial distance r0 D 0.05, and the lower curve 2 to r0 D 0.005. For convenience, we also show the straight line 3, which represents the cubic law 2 .t / / t 3 . 1000

1000

100

100

10

10

1 0.1 0.01 0.01

1 3

1

0.1

2

0.01 0.1

1 Time t

10

100

0.01

3

1 2 0.1

1 Time t

10

100

Figure 7.4. Left panel: The function .t / D hr 2 .t /i1=2 for two different initial distances: r0 D 0.05 (curve 1) and r0 D 0.005 (curve 2); C0 D 4, k0 D 0.01. Curve 3 represents the cubic law 2 / t 3 . Right panel: the same curves, but for C0 D 6.

Section 7.3 Calculations by the combined Eulerian–Lagrangian stochastic model

151

1,000 100 10 .t / 1 3 0.1 0.01 0.01

1 2 0.1

1 10 Time t

100

Figure 7.5. The same as in Figure 7.4, for C0 D 6 and k0 D 0.001.

From these curves it is clearly seen that the Richardson cubic law hr 2 .t /i D g "t N 3 with g D 6.65 for C0 D 4 and g D 2.92 for C0 D 6 is satisfied with high accuracy in the range of about two decades (see the relevant intervals where the curves are well approximated by linear dependencies in Figure 7.4). In contrast to the power laws discussed in Section 7.2 (see Figures 7.1–7.3) the value of the constant g in .7.1/ does not change (to within 3 % of accuracy) when the value of k0 varies. Indeed, when k0 D 0.001, we find from Figure 7.5 that g D 2.96. The value of C0 , the number of harmonics, the number of trajectories, and the initial distances were the same as in Figure 7.4. In addition, the inertial subrange is extended, as expected, when increasing the integral scale of turbulence. Finally we note that the stabilization of numerical results happens when the number of harmonics in .7.6/ reaches the value 10: further increase does not noticeably change the value of .t /. Remark 7.1. Thus our combined Eulerian–Lagrangian stochastic model confirms the Richardson cubic law .7.1/. The constant g in this law essentially depends on the universal constant C0 . The calculations presented above suggest that g ' 2.9 for C0 D 6 and g ' 6.6 for C0 D 4. These values agree with the results given by Kraichnan [101] (g D 2.42), Lundgren [135] (g D 3), and Larcheveque and Lesieur [125] (g D 3.5). In [149] the effective Hamiltonian method was used to derive the Richardson cubic law. From this approach it is possible to estimate that g varies from 3.5 to 9 when the decorrelation parameter a changes from 2 to 0.5. Thomson’s Lagrangian model [238] suggests that g varies between 1 and 2 when C0 varies between 4 and 6. Values close to these were obtained by Borgas and Sawford [15].

152

Chapter 7 Stochastic Lagrangian models for 2-particle motion in turbulent flows

7.3.2 Thomson’s “two-to-one” reduction principle We recall the following obvious relation: Z P1L .x, vE, t ; x0 / D P2L .x, vE, x0 , vE0 , t ; x0 , x00 /d x0 d vE0 ,

(7.15)

where P1L , P2L are the 1- and 2-particle Lagrangian p. d. f.’s: v  V.t , x0 //i, P1L .x, vE, t ; x0 / D hı.x  X.t , x0 //ı.E and v  V.t , x0 // P2L .x, vE, x0 , vE0 , t ; x0 , x00 / D hı.x  X.t , x0 //ı.E ı.x0  X.t , x00 //ı.E v 0  V.t , x00 //i. Here we denote for simplicity X.t , x0 / D X.t ; x0 , t0 /,

V.t , x0 / D V.t ; x0 , t0 / D UE .X.t , x0 /, t /.

However, in the Lagrangian models this relation is generally not satisfied. Therefore, in [238] it was noted that when constructing 2-particle Lagrangian models, it is desired to satisfy the relation .7.15/. This principle is called in [238] a “two-to-one” reduction. In [15], some new criteria were derived from Thomson’s “two-to-one” reduction. As far as we know, there is no Lagrangian model exactly satisfying this principle. It is difficult to verify the relation .7.15/ for a particular Lagrangian model. Therefore, more simple consequences of this relation can be studied. For instance, the 1-particle mean square velocity component  2 .t / D 13 hjV.t /j2 i was calculated in [15]. Note that in our case of stationary incompressible isotropic turbulence,  2 is a constant equal to the Eulerian 1-point mean square velocity component 13 hjUE j2 i. If a 2-particle stochastic model satisfies this property, we say that the “two-to-one” reduction principle in a broad sense is satisfied. Note that there is no Lagrangian model exactly satisfying the “two-to-one” reduction principle even in this broad sense. In Figure 7.6 we plot the mean square Lagrangian velocities for both particles i2 .t / D 13 hV2i .t /i, i D 1, 2, at C0 D 6, k0 D 1, kmax D 104 , and for the initial separation r0 D 103 which is much less than the characteristic spatial scale L D 2. From these curves we see that the “two-to-one” reduction principle in a broad sense is approximately satisfied. Note that  2 .t / deviates from the true value  D 1.4 to within 7 %, for statistics about N D 4,000. When increasing the accuracy (i. e., N D 50,000), it turns out that in the initial time interval .0, 0.5/ the deviation from the constant remains the same, while it decreases to  1% for t  0.5 (see Figure 7.6, right panel). Note that the characteristic external time scale of turbulence estimated as L2=3 =N"1=3 D .2/2=3 . Hence, in our calculations, the time interval where we see the 7 % deviation is much less than this external time scale.

Section 7.3 Calculations by the combined Eulerian–Lagrangian stochastic model 2 1.8 1.6 i2 .t / 1.4 1.2 1 0.8 0.6 0.4 0.2 0

0 0.5 1 1.5 2 2.5 3 3.5 4

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

0 0.5 1 1.5 2 2.5 3 3.5 4

Time t

Time t

Figure 7.6. The functions D N D 50,000 (right panel) samples. i2 .t /

153

1 hV2i .t /i, 3

i D 1, 2, for N D 4,000 (left panel) and

Note that the behavior of this deviation is not surprising. Indeed, for small times, when the instantaneous distance r.t / is less than ˛L, as seen from the definition of the function  , the motion equation is dR D †V, dt where

dr D vE, dt

(7.16)

² ³   1 r.t / r.t / UE R.t / C , t C UE R.t /  ,t . †V D 2 2 2

Therefore, the Lagrangian velocities of the first and second particles can be written as

vE V1 D †V  , 2

vE V2 D †V C , 2

respectively. By definition, for small times, †V and vE are independent, and hj†V.t /j2 i  hjUE j2 i. We conclude from this that both particles have a small contribution of the order hjE v .t /j2 i to their mean square Lagrangian velocities. Since 2 hjE v .t /j i increases linearly in time, these arguments explain why the curves in Figure 7.6 slightly increase at the beginning. For times when r.t / > ˇL, our equation of motion of two particles coincides with the equation of motion in the pseudoturbulence UE which results in Figure 7.6 in the small deviation of the curves from the exact constant. When choosing the parameters ˛ and ˇ we have to balance between two situations: on the one hand, taking smaller values of ˛ and ˇ, we ensure that the “two-to-one” reduction principle will be better satisfied. On the other hand, for a highReynolds-number turbulence, the physically natural model .7.16/ is reasonable for r.t /=L  ˛0 , with ˛0 being a constant whose value depends on the velocity features. Therefore, if we take the value of ˛ too small, we ignore this physical picture.

154

Chapter 7 Stochastic Lagrangian models for 2-particle motion in turbulent flows

In all the above considerations, we assumed that the initial distance between two particles is small compared to the characteristic spatial scale of turbulence. This is the worst case, and for larger initial distances, the “two-to-one” reduction principle is satisfied with even higher accuracy. Moreover, these arguments make it clear that for times .t / > ˇL the “two-to-one” reduction principle will be satisfied in our model even in the strong formulation .7.15/.

7.3.3 Concentration fluctuations As we have seen from the calculations of the function .t /, the pseudoturbulent model and the combined Eulerian–Lagrangian stochastic model lead to qualitatively different results in the inertial subrange. Another important quantity which is commonly considered as a fine test (e. g., see [44, 214, 238]) is the intensity of the concentration fluctuations of particles in a fixed point x defined as p Ic .x, t / D

hc 2 .x, t /i  hc.x, t /i2 , hc.x, t /i

where c.x, t / is the instantaneous concentration at time t in the point x. In this section we use this test to compare the pseudoturbulent model and the combined Eulerian– Lagrangian stochastic model. We take an instantaneous deterministic source of particles in the Gaussian form S.x/ D

² ³ jxj2 1 exp  , 202 .2/3=2 03

(7.17)

where 0 is the characteristic size of the puff. The mean and the second moment of concentration can be represented as [146] Z hc.x, t /i D p1L .x, t ; x0 , t0 /S.x0 /d x0 , Z hc.x, t /c.x0 , t /i D p2L .x, x0 , t ; x0 , x00 , t0 /S.x0 /S.x00 /d x0 d x00 . The functions p1L and p2L are the 1-point and 2-point p. d. f.’s: p1L .x, t ; x0 , t0 / D hı.x  X.t ; x0 , t0 //i, p2L .x, x0 , t ; x0 , x00 , t0 / D hı.x  X.t ; x0 , t0 //ı.x0  X.t ; x00 , t0 //i, where X.t ; x0 , t0 / is the Lagrangian trajectory started at time t0 < t in the point x0 . We call this path a direct Lagrangian trajectory. Backward Lagrangian trajectory starting

Section 7.3 Calculations by the combined Eulerian–Lagrangian stochastic model

155

at t at the point x is defined as X.t0 ; x, t /, the solution of the equation @X D UE .X, t0 /, @t0

t0 < t

with the initial condition X.t / D x. Note that in the incompressible velocity field the direct and backward p. d. f.’s are equal (e. g., see [135]): p1L .x, t ; x0 , t0 / D p1L .x0 , t0 ; x, t /, p2L .x, x0 , t ; x0 , x00 , t0 / D p2L .x0 , x00 , t0 ; x, x0 , t /. These relations are the basis for the backward trajectory technique (e. g., see [214]) which uses the equalities hc.x, t /i D hS.X.t0 ; x, t //i, hc 2 .x, t /i D hS.X.t0 ; x1 , t //S.X.t0 ; x2 , t //i. The latter relation involves two backward trajectories, X.t0 ; x1 , t / and X.t0 ; x2 , t / and the equality is understood in the sense that in the right-hand side the limit is taken as x2 , x1 ! x with the rate satisfying jx2  x1 j /  (e. g., see [44]). This approach is convenient when it is necessary to calculate the concentrations at a fixed point provided the source has relatively large spatial extension. Thus to calculate the mean of c.x, t /, we start at the point x at the time t , and simulate the Lagrangian trajectory backward in time till the time instant t0 , then calculate the random estimator S.X.t0 ; x, t //. After averaging over large number of trajectories we get the mean concentration. The same is true for hc 2 .x, t /i: we start two backward trajectories at points x1 and x2 at the time t separated by very small distance (which is of the order of the Kolmogorov inner scale ), and calculate the random estimator S.X.t0 ; x1 , t // S.X.t0 ; x2 , t //. After averaging over a large number of pairs of trajectories we get the second moment of the concentration. We used the backward trajectory technique to calculate the intensity of concentration fluctuations for the source .7.17/ with 0 D 0.1u30 =N", where 3u20 is the energy of turbulence (see .7.5/). N 0 /" In Figure 7.7, we show Ic .0, t / as a function of the dimensionless time  D .tt u20 in log-log coordinates. The upper curve (2) is obtained by the pseudoturbulent model where N D 16,000 pairs of trajectories were used, and the lower curve (1) is obtained by the combined Eulerian–Lagrangian stochastic model with N D 30,000 pairs. In the latter case, C0 D 6, ˛ D 0.05, ˇ D 0.1. In both cases, the wave interval from k0 D 2 to kmax D 103 was used. In the pseudoturbulent model 50 harmonics were needed, while in the combined model 30 harmonics were sufficient to give stable numerical results.

156

Chapter 7 Stochastic Lagrangian models for 2-particle motion in turbulent flows 10 2

1

1 Ic .0, t / 0.1

0.01 0.01

0.1



1

10

Figure 7.7. Intensity of concentration fluctuations as a function of dimensionless time  D .t t0 /"=u N 20 . Upper curve 2: the pseudoturbulent model (N D 16,000 samples); lower curve 1: the combined Eulerian–Lagrangian stochastic model (N D 30,000 samples).

Calculations of Ic .0, t / by both models are of relatively high accuracy in the interval .0.01, 1/. Then the error increases because of the large variance of the second moment hc 2 i. Note that the mean concentration is calculated with small error in all the interval  2 .0.01, 10/. Both models give close results for Ic .0, t / in the interval  2 .0.01, 0.1/. Further the difference between the results of two models is increasing to about of 500 %. This shows that the combined Eulerian–Lagrangian stochastic model qualitatively differs from the pseudoturbulent stochastic model. Note that the combined Eulerian–Lagrangian stochastic model agrees fairly well with the results reported by Thomson [238].

7.4 Technical remarks In this section we give some details of the numerical schemes applied both to the pseudoturbulent stochastic model described in Section 7.2, and to the combined Eulerian– Lagrangian stochastic model of Section 7.3. Theoretically, we are dealing with the equation of motion in a Gaussian random velocity field with a given continuous spectral tensor. However, numerically we are dealing with some approximate random field where only a finite number of harmonics is involved. Relevant convergence results can be found in [143]. In practice, we increase the number of harmonics until the numerical results become stable. It should be noted that for the pseudoturbulent stochastic model of Section 7.1, when calculat-

Section 7.4 Technical remarks

157

ing the mean square separation, it was sufficient to take 25 harmonics per decade of the interval .k0 , kmax / where the energy spectrum is located. Then the total number O Note that the model of harmonics in this model approximately equals 25 log10 .Re/. 3=4 O of [62,63] requires / .Re/ harmonics. In our calculations the cost of the combined Eulerian–Lagrangian stochastic model is much lower: the total number of harmonics providing stable numerical results for the mean square separation was about 10. In the case of the intensity of concentration fluctuations, the needed number of harmonics was about 30. Another important issue of the numerical scheme of integration of the motion equation is the choice of the time step of discretization. We used the standard Euler scheme X.t C t / D X.t / C t UE .X.t /, t /

(7.18)

for the pseudoturbulent stochastic model, and R.t C t / D R.t / C t †V, ² 

  ³ r.t / r.t / r.t C t / D r.t / C t  V C 1   vE.t / , L L p vE.t C t / D vE.t / C a.r.t /, vE.t //t C 2C0 "Nt  t ,

(7.19)

² ³   1 r.t / r.t / UE R.t / C , t C UE R.t /  ,t , 2 2 2   r.t / r.t / , t  UE R.t /  ,t , V D UE R.t / C 2 2 and  t are standard 3D Gaussian vectors mutually independent at different time instances. We choose a variable time step  2 1=3 r .t / t D ı , "N where

†V D

where ı is a small adjusting parameter. This choice was also used in [238]. The choice of the parameter ı in the scheme .7.18/ should be consistent with the initial distance r0 . In our case, for r0 D 0.01 it was sufficient to take ı D 0.05. When r0 D 104 , it requires a much smaller time step. However, we took ı D 0.05 which cannot be considered sufficient and could be further diminished at a small initial time interval. We have actually carried out such high-accuracy calculations with  D 0.01 and r0 D 104 for small time intervals (t  0.1). From these calculations we conclude that the parameter in .7.2/ cannot be less than that given in Section 7.1. In .7.19/ we put ı D 0.01. These values of ı ensure stable numerical results to within the statistical error of the Monte Carlo calculations.

158

Chapter 7 Stochastic Lagrangian models for 2-particle motion in turbulent flows

7.5 Conclusion A 2-particle combined Eulerian–Lagrangian stochastic model which correctly reflects the behavior in the inertial subrange was developed. In this chapter, which develops the results described in the previous chapter, we presented the calculation details, which confirm that the classical pseudoturbulence model does not mimick the Richardson law (at least for the model Reynolds numbers about 109 ), while the combined model O  106 . However reproduces the Richardson law with high accuracy already for Re the classical pseudoturbulence model leads to a stable power law .7.2/, which depends on the parameter "N and the characteristic external spatial scale of turbulence. The new combined model manifests the “two-to-one” reduction principle. The cost of the Monte Carlo technique proposed for simulation of the pseudoturbulent dispersion depends logarithmically on the model Reynolds number. The combined Eulerian–Lagrangian stochastic model is even more efficient, since the variable time step of integration increases in this model faster, compared to the case of the pseudoturbulent stochastic model. In the combined Eulerian–Lagrangian stochastic model two parameters, ˛ and ˇ, are introduced which have clear physical meaning: when r.t / < ˛L, the processes of relative and absolute motions are approximately statistically independent, while for r.t / > ˇL, these processes are strongly correlated. We adjusted these parameters from the “two-to-one” reduction principles. Generally, more physics should be taken into account to make a proper choice of ˛ and ˇ. In conclusion we note that the combined Eulerian–Lagrangian stochastic model presented for classical pseudoturbulence can be easily extended to more general turbulent flows. Indeed, having models with a given large-scale statistical structure UE , say, obtained by DNS or large eddy simulation methods, we can put it into the model .7.10/.

Chapter 8

The 1-particle stochastic Lagrangian model for turbulent dispersion in horizontally homogeneous turbulence

A 1-particle stochastic Lagrangian model in 2 and 3 dimensions is constructed for transport of particles in horizontaly homogeneous turbulent flows with arbitrary 1-point probability density function. It is shown that in the case of anisotropic turbulence with Gaussian p. d. f., this model essentially differs from the known Thomson model. The results of calculations according to our model in the case of neutrallystratified atmospheric surface layer agree satisfactorily with the measurements known from the literature.

8.1

Introduction

This chapter deals with 1-particle stochastic Lagrangian models for 2D and 3D turbulent transport. Here we treat the fully developed turbulence (i. e., a flow with a very high Reynolds number) as a random velocity field .u, v, w/ which is assumed to be incompressible. Therefore, the trajectories of particles in such flows are stochastic processes. To simulate these stochastic processes, two different approaches are known in the literature. The first one is based on the numerical solution of the system of random equations @X D u.X , Y , Z, t /, @t @Y D v.X , Y , Z, t /, @t @Z D w.X , Y , Z, t /. @t

(8.1)

Here X.t /, Y .t /, Z.t / are the coordinates of the Lagrangian trajectory at time t . The random fields u, v, w are simulated by Monte Carlo methods (e. g., see [42, 63, 102, 191,198,244]), and the random trajectories are then obtained by the numerical solution of .8.1/ with the relevant initial data.

160

Chapter 8 The 1-particle stochastic Lagrangian model

In the second approach the true trajectory X.t /, Y .t /, Z.t / is assumed to be approxO /, a solution to a stochastic differential imated by a model trajectory XO .t /, YO .t /, Z.t equation of the Ito type (e. g., see [215, 237]: d XO D UO dt , d YO D VO dt , d ZO D WO dt , O YO , Z, O UO , VO , WO /dt C bu .t , X, O YO , Z, O UO , VO , WO / dBu .t /, d UO D au .t , X, O YO , Z, O UO , VO , WO /dt C bv .t , X, O YO , Z, O UO , VO , WO / dBv .t /, d VO D av .t , X, O YO , Z, O UO , VO , WO /dt C bw .t , X, O YO , Z, O UO , VO , WO / dBw .t /. (8.2) d WO D aw .t , X, Here UO , VO , WO are the components of the model Lagrangian velocity, Bu .t /, Bv .t /, Bw .t / are three standard independent Wiener processes. Ideally, one would have an approximation such that the true and model Lagrangian velocities coincide: O /, t /, UO .t / D u.XO .t /, YO .t /, Z.t O /, t /, VO .t / D v.XO .t /, YO .t /, Z.t O /, t /, WO .t / D w.XO .t /, YO .t /, Z.t

(8.3)

which would assure that the true and model trajectories are the same. However it is unrealistic to satisfy .8.3/; therefore one uses different consistency principles. Namely, the general consistency principle says that the statistics of the model process O /, UO .t /, VO .t /, WO .t / XO .t /, YO .t /, Z.t satisfies the same relations which are satisfied by the true process X.t /, Y .t /, Z.t /, U.t /, V .t /, W .t /, where U.t / D u.X.t /, Y .t /, Z.t /, t /, V .t / D v.X.t /, Y .t /, Z.t /, t /, and W .t / D w.X.t /, Y .t /, Z.t /, t / are the components of the true Lagrangian velocity. Two consistency criteria used in the literature are: (a) consistency with the Kolmogorov similarity theory; (b) consistency with Novikov’s integral relation. Here (a) reads h.d U /2 i D h.d V /2 i D h.d W /2 i D C0 "dt , and hd U d V i D hd U d W i D hd W d V i D 0, where d U , d V , d W are the components of the increments of the Lagrangian velocity, " is the mean rate of the dissipation of turbulence energy, C0 is the universal constant (e. g., see [146, 215, 237]); here, and in the following, the angle brackets stand for the ensemble average over the samples of the random velocity field.

161

Section 8.1 Introduction

Note p that (A) implies (e. g., see [237]) that in .8.2/, all the terms bu , bv , bw are equal to C0 ": p (8.4) bu D bv D bw D C0 ". Novikov’s integral relation has the form [153] Z pE .u, v, w; x, y, z, t / D pL .x, y, z, u, v, w; x0 , y0 , z0 , t /dx0 dy0 dz0 .

(8.5)

R3

Here pE is the probability density function (pdf) of the Eulerian velocity u, v, w in the fixed point x, y, z at the time t , and pL is the joint p. d. f. of the true Lagrangian phase point .X , Y , Z, U , V , W / defined by the trajecory started at x0 , y0 , z0 . Thus the consistency with the Novikov relation .8.5/ means that the p. d. f. of the model phase point governed by .8.2/, say pOL , satisfies Z pOL .x, y, z, u, v, w; x0 , y0 , z0 , t /dx0 dy0 dz0 . (8.6) pE .u, v, w; x, y, z, t / D R3

Note that .8.6/, the Focker–Planck–Kolmogorov equation for pOL and .8.4/ lead to the well-mixed condition due to D. Thomson [237]: @ @ @pE @pE @pE @ @pE Cu Cv Cw C .au pE / C .av pE / C .aw pE / @t @x @y @z @u @v @w ² ³ @2 pE @ 2 pE C 0 " @2 pE C C D . (8.7) 2 @u2 @v 2 @w 2 In this chapter we study a horizontally-homogeneous turbulent flow which implies that pE does not depend on x, y. Therefore, in the left-hand side of .8.7/ the second and third terms vanish. Here the main problem is that .8.7/ does not uniquely define the coefficients au , av , and aw of the model .8.2/. Indeed, even for homogeneous turbulence, in [217] two different choices of au , av , aw are presented, both satisfying the well-mixed condition .8.7/, but whose statistical characteristics are different. For the Gaussian form of pE , one of the appropriate techniques for getting the coefficients au , av , aw is given in [237]. In the non-Gaussian 3D case, to the authors knowledge there is no appropriate choice of these coefficients. In 2D a non-Gaussian case was treated by Flesch and Wilson in [56]. These authors mentioned that the two different models do not lead to essentially different results in the case of Gaussian pE . As reported in [56], the same is true for two models considered in [217]. In this chapter we suggest a proper choice of the coefficients au , av , aw in a general case of p. d. f. pE . Our derivation is based on some assumptions which ensure a unique choice of the model. It should be stressed that even in the Gaussian case our model essentially differs, as shown below (Section 8.4), from the model given by Thomson [237]. This confirms our opinion that it is necessary along theoretical studies to extract additional information from experiments.

162

Chapter 8 The 1-particle stochastic Lagrangian model

8.2 Choice of the coefficients in the Ito equation Let us formulate the main assumptions about the Lagrangian model of the type .8.2/. We consider a horizontally homogeneous incompressible high-Reynolds-number turbulent flow in the space R3 . Thus the mean velocity has no vertical component. Without loss of generality we assume in addition that the mean velocity is directed along the x-axis. Thus the mean velocity vector is .u.x, N y, z, t /, 0, 0/, while pE and uN do not depend on x, y. We will write the p. d. f. pE in the form 0 .u0 , v 0 , w 0 ; z, t / pE .u, v, w; z, t / D pE

N t /, v 0 D v and w 0 D w. where u0 D u  u.z, By .8.4/, equation .8.2/ in these variables has the form O t //dt , d YO D VO 0 dt , d ZO D WO 0 dt , N Z, d XO D .UO 0 C u. p O UO 0 , VO 0 , WO 0 /dt C C0 " dBu .t /, d UO 0 D au0 .t , Z, p O UO 0 , VO 0 , WO 0 /dt C C0 " dBv .t /, d VO 0 D av0 .t , Z, p 0 O UO 0 , VO 0 , WO 0 /dt C C0 " dBw .t /. d WO 0 D aw .t , Z,

(8.8)

The well-mixed condition in new variables is 0 @pE @p 0 @ @ @ 0 0 / C 0 .av0 pE /C .a0 p 0 / C w 0 E C 0 .au0 pE @t @z @u @v @w 0 w E ² 0 0 0 ³ @2 pE @2 pE C 0 " @2 pE D C C . 2 @.u0 /2 @.v 0 /2 @.w 0 /2

(8.9)

0 does Assumption 8.1. We assume in addition that au0 does not depend on v 0 while aw 0 0 0 0 0 0 0 0 0 not depend on u , v , i. e., au D au .t , z, u , w /, aw D aw .t , z, w /.

Then the model .8.8/ reads O t //dt , d YO D VO 0 dt , d ZO D WO 0 dt , d XO D .UO 0 C u. N Z, p O UO 0 , WO 0 /dt C C0 " dBu .t /, d UO 0 D au0 .t , Z, p O UO 0 , VO 0 , WO 0 /dt C C0 " dBv .t /, d VO 0 D av0 .t , Z, p 0 O WO 0 /dt C C0 " dBw .t /. d WO 0 D aw .t , Z,

(8.10)

Integrating .8.9/ over u0 and v 0 yields 0 0 @p1E @p 0 C0 " @2 p1E @ 0 0 0 .a .t , z, w /p / D , C w 0 1E C w 1E @t @z @w 0 2 @.w 0 /2

(8.11)

Section 8.2 Choice of the coefficients in the Ito equation

163

where 0 p1E

D

0 p1E .w 0 ; z, t /

Z D

Z

1

1

1

0 pE .u0 , v 0 , w 0 ; z, t / du0 dv 0 .

1

(8.12)

Here we have assumed that 0 0 au0 pE , av0 pE ,

0 0 @pE @pE , @u0 @v 0

all tend to zero as .u0 /2 C .v 0 /2 ! 1.

Similarly, the integration of .8.9/ over v 0 leads to 0 @p 0 @p2E @ @ 0 0 /C .a0 .t , z, w 0 /p2E / C w 0 2E C 0 .au0 .t , z, u0 , w 0 /p2E @t @z @u @w 0 w  2 0 0 @2 p2E C0 " @ p2E D C , 2 @.u0 /2 @.w 0 /2

where 0 p2E

D

Z

0 p2E .u0 , w 0 ; z, t /

D

1

1

0 pE .u0 , v 0 , w 0 ; z, t / dv 0 .

(8.13)

(8.14)

Now, under the assumption made about the behavior at infinity, it is possible to unique0 . Indeed, from .8.11/ one gets a0 , then from ly define the coefficients au0 , av0 and aw w 0 .8.13/ one finds au , and from .8.9/ one obtains av0 . This yields 0 aw .t , z, w/ D

²  ³ 0 1 C0 " @p1E @f1E @F1E  C , 0 .w; z, t / p1E 2 @w @t @z

(8.15)

where Z f1E .w; z, t / D

w

1 w

Z F1E .w; z, t / D

1

0 p1E .w 0 ; z, t / dw 0 , 0 w 0 p1E .w 0 ; z, t / dw 0 ,

and au0 .t , z, u, w/ ² 0 ³ @2 f2E   @f2E @f2E  @  0 1 C0 "  @p2E  C Cw  a f2E , (8.16) D 0 p2E 2 @u @w 2 @t @z @w w where

Z f2E .u, w; z, t / D

u

1

0 p2E .u0 , w; z, t / du0 .

164

Chapter 8 The 1-particle stochastic Lagrangian model

Finally, av0 .t , z, u, w/

  0 @pE @ 2 fE @fE C 0 " @ 2 fE @fE C Cw D C  2 @u2 @v @w 2 @t @z ³   @ @  .au0 fE /  a 0 fE , (8.17) @u @w w ²

1 0 pE

where

Z fE .u, v, w; z, t / D

v 1

0 pE .u, v 0 , w; z, t / dv 0 ,

Thus the coefficients .8.15/–.8.17/ define a unique stochastic model .8.10/ through 0 . the p. d. f. pE Remark 8.1. This model is a natural extension of the 1-dimensional (in z direction) Thomson model [237] in the sense that the vertical coordinates .z, w/ are governed in our model by a SDE which coincides with the Thomson model. Note that in Thom0 ) this is not the case: the statistics of .z, w/ in son’s 3D model (with Gaussian pE his 3D model essentially differ from those of .z, w/ in his 1-dimensional model (see Section 8.4).

8.3 2D stochastic model with Gaussian p. d. f. In this section we present concrete expressions for the coefficients in the case of Gaussian p. d. f.. We extract a 2D model from a 3D model as O t //dt , d ZO D WO 0 dt , N Z, d XO D .UO 0 C u. p O UO 0 , WO 0 /dt C C0 " dBu .t /, d UO 0 D au0 .t , Z, p 0 O WO 0 /dt C C0 " dBw .t /. d WO 0 D aw .t , Z, In the Gaussian case 0 .u, w; z, t / p2E

´

μ w2 D exp  2 .u  /  2 , 2u=w w 2u=w 2w 1

1

2

(8.18)

(8.19)

where u=w D

1=2 , w

D

uw w, w2

 D u2 w2  .uw/2 ,

and u2 , w2 are the variances of the x- and z-velocity components, respectively. From .8.19/ ² ³ 1 w2 0 .w; z, t / D p exp  2 , p1E (8.20) 2w 2w

165

Section 8.3 2D stochastic model with Gaussian p. d. f.

then,

Z f1E .w; z, t / D

 w 1 2 p exp .t =2/ dt D ˆ , w 2

w w

1

0 .w; z, t /. F1E .w; z, t / D w2 p1E

Note that

0 1 @p1E w D 2, 0 p1E @w w

and



1 @F1E @w2 1 2 C 1/ D .w 0 p1E @z 2 @z

w @w P @f1E D 2 ˆ.w=w /, @t w @t

P /D where ˆ.

dˆ , d

and Z ˆ. / D



1

1 p exp .t 2 =2/ dt . 2

From .8.15/ we find 0 .t , z, w/ D  aw



C0 " 1 @w  2w2 w @t

wC

1 @w2 2 @z



w2 C 1 . w2

(8.21)

Note that this coincides with Thomson’s relevant expression in his 1D model [237]. By the definition,  u

0 f2E .u, w, z, t / D p1E .w; z, t / ˆ . u=w To find au0 from .8.16/, we need the expressions for @f2E , @t By definition we get

@f2E , @z

@f2E , @w

0 @p2E , @u

@2 f2E . @w 2

 ³ ² 0 @p2E @ .u  / 0 @f2E 1 @w2 w 2 p  1 C ‰./ D 2 D f2E , @u u=w 2E @t 2w2 @t w2 @t  ³ ² @f2E @ 1 @w2 w 2  1 C ‰./ D f2E , @z 2w2 @z w2 @z ³ ² @f2E w (8.22) D f2E  2  ‰./ , @w w ´ μ i2 h 1 @2 f2E w P D f2E  2  ‰./  2 C 2 ‰./ , @w 2 w w

166

Chapter 8 The 1-particle stochastic Lagrangian model

where ‰. / D

d ln ˆ. /, d

P / D d ‰. / , ‰. d

D

u

, u=w

D

uw . u=w w2

Substituting .8.22/ in .8.16/ yields C0 " .u  / 2 2u=w ² ³ 0 @f2E C0 " @2 f2E 1 @aw @f2E 0 @f2E w  f2E  aw C C 0  p2E @t @z @w @w 2 @w 2  ² f2E @ C0 " 1 @w2 w 2  1  ‰./ D  2 .u  / C 0 (8.23)  2 2u=w p2E 2w @t w2 @t   



0 @ @aw 1 @w2 w 2 w 0  1 C ‰./   aw  2  ‰./ w 2w2 @z w2 @z @w w 2

 ³ C0 " 1 w P C  2  ‰./  2 C 2 ‰./ . 2 w w

au0 D 

Since

f2E ‰ D u=w , 0 p2E

we find from .8.23/ au0 .t , z, u, w/

P ‰./ D ‰./. C ‰.// ,

 C0 ".1 C 2 /  @w2 D .u  w/ C 2 C0 " C w 2 2u=w 2w @t    @w2 w 2 @ @ C C 1  u=w Cw . 2 @z w2 @t @z

(8.24)

Here

u  w uw , D . 2 w u=w Note that in the stationary case these expressions can be simplified to  C0 " 1 @w2 w 2 0 aw .t , z, w/ D  2 w C C1 , 2w 2 @z w2 D

C0 ".1 C 2 /  C0 " .u  w/ C w 2 2u=w 2w2   @w2 w 2 @ C C 1  u=w w . 2 @z w2 @z

au0 .t , z, u, w/ D 

(8.25)

In the next section we present some numerical experiments which show that the model presented essentially differs from Thomson’s model [237].

167

Section 8.4 Numerical experiments

8.4

Numerical experiments

In this section we compare our model against Thomson’s model in the case of a 2D stationary turbulence. First we consider the case of a 2D homogeneous turbulence specified by (8.26) u D bu u , w D bw u , where u is defined by u2 D uw; bu and bw are some dimensionless constants. The mean velocity field is zero. Thomson’s model in the stationary homogeneous case reads [56, 237] d XO D UO 0 dt , d ZO D WO 0 dt , p d UO 0 D au0 .UO 0 , WO 0 /dt C C0 " dBu .t /, p 0 d WO 0 D aw .UO 0 , WO 0 /dt C C0 " dBw .t /

(8.27) (8.28)

with au0 .u, w/ D 

C0 " 2 . u C u2 w/, 2 w

0 aw .u, w/ D 

C0 " 2 . w C u2 u/. 2 u

(8.29)

where  D u2 w2  u4 . We calculated the dimensionless vertical eddy diffusivity k. / D

"hZO WO i , u4

(8.30)

2 2

where  D t =TL . Here TL D C0w" is the Lagrangian time scale in z-direction. Since this characteristic depends only on z, w, it is sufficient to take in our model .8.18/ only the equation governing z, w: d ZO D WO 0 dt , p 0 d WO 0 D aw .UO 0 , WO 0 /dt C C0 " dBw .t /, where 0 aw .U , W / D 

(8.31)

C0 " W. 2w2

We note again (see Remark 2.1) that .8.31/ is exactly the Thomson 1D model [237]. In our numerical experiments, we have fixed bu D 2.3, and have made calculations for bw D 0.8, 1, 1.2, and 1.4. These values characterize the anisotropy in the neutrallystratified surface layer of the atmosphere [19]. The values of C0 and " were taken as 4 and 1 m2 /s3 , respectively, while u D 0.4 m/s. In Figure 8.1 we show the function k. / (see .8.30/) obtained by Thomson’s 2D model .8.27/ (solid curves) and Thomson’s 1D model .8.31/ (dashed curve). The

168

Chapter 8 The 1-particle stochastic Lagrangian model Eddy diffusivity k. / 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 a 0.24 C 2

2

22

22

22

2

22

4 2 444 4 2 44444 2 4 CCCCCCCCC 2 44 4 CCC C C 4 ?????????????? 2 C ? C a ? 2 ?a 4 a a a a a a a a a a a a a a a C ?a 4 C 2 a? 4 C2 4 2 2

4

bw D 0.8

bw D 1 bw D 1.2 bw D 1.4 Thomson’s 1D

6 8 10 12 14 16 18 20 Dimensionless time 

Figure 8.1. Dimensionless vertical eddy diffusivity coefficient as a function of  , for different values of bw .

lower solid line corresponds to bw D 1.4, the next one to bw D 1.2, the next to bw D 1.0, and the very upper curve to bw D 0.8. It is clearly seen that the diffusivity coefficients of this two models essentially differ, e. g., for bw D 0.8, this difference is about a factor of 3 at the steady-state values of k. /. With decreasing anisotropy (i. e., the ratio bu =bw decreases), this difference becomes smaller. Thus we conclude that even in the homogeneous turbulence (which is, however, anisotropic) the two studied models may give essentially different results. To choose a proper case, one would need relevant measurements. We have no such experimental results in the homogeneous case, while in the neutrally-stratified surface layer (NSSL) the measurements are at hand (e. g., see [19]); therefore, it is interesting to compare the 2D Thomson model with our model defined by .8.18/, .8.25/. Thomson’s 2D model of 1-particle dispersion in horizontally homogeneous stationary turbulent flow reads [56, 237] O t //dt , d ZO D WO 0 dt , N Z, d XO D .UO 0 C u. p O UO 0 , WO 0 /dt C C0 " dBu .t /, d UO 0 D au0 .Z, p 0 O UO 0 , WO 0 /dt C C0 " dBw .t /, .Z, d WO 0 D aw

(8.32)

169

Section 8.4 Numerical experiments

where  1 d uw C0 "  2 w u  uww C 2 2 dz ² ³  d uw   w du2  2 C w u  uww C  uwu C u2 w , 2 dz dz

au0 .z, u, w/ D 

 1 dw2 C0 "  2 u w  uwu C 2 2 dz ² ³  dw2   w d uw  2 2 w u  uww C  uwu C u w . C 2 dz dz

0 .z, u, w/ D  aw

Here  D u2 w2  uw 2 . For the NSSL, the coefficients in this model can be taken as follows [19, 146]: ".z/ D

u3 , z

u.z/ N D

u ln .z=z0 /,

and u and w are given by .8.26/ with u2 D uw Dconst; D 0.4, z0 is the roughness height. Hence, the Thomson model in this case is specified by au0 .z, u, w/ D 

 C0 ".z/  2 w u C u2 w , 2

0 aw .z, u, w/ D 

 C0 ".z/  2 u w C u2 u . 2

Our model .8.18/, .8.25/ in the case of NSSL is determined by au0 .z, u, w/ D  0 .z, w/ D  aw

C0 ".z/.1 C 2 /  C0 ".z/ .u  w/ C w, 2 2u=w 2w2 C0 ".z/ w. 2w2

(8.33)

In the comparative calculations, we have calculated the following dimensionless Lagrangian characteristics: p hZ 2 .t /i hZ.t /i A.t / D , B.t / D , u t u t ² ³ z0 hX.t /i C.t / D exp C1 . (8.34) u t u t It is known (e. g., see [19, p. 77]) that these functions tend, as t ! 1, to constant values a, b and c, respectively, provided hs and z0 are much less than u t . Here hs is the height at which the Lagrangian trajectory starts. The experimental measurements of the constants a, b, and c are scattered to a certain amount. However as can be

170

Chapter 8 The 1-particle stochastic Lagrangian model

Table 8.1. Steady-state values of the functions .8.34/, for bw D 1.2. Model Thomson’s ours

a

b

c

0.65˙ 0.01 0.52˙ 0.02

0.48˙ 0.02 0.38˙ 0.01

0.21˙ 0.03 0.14˙ 0.01

Table 8.2. Steady-state values of the functions .8.34/, for bw D 1.3. Model Thomson’s ours

a

b

c

0.78˙ 0.01 0.66˙ 0.02

0.59˙ 0.02 0.49˙ 0.02

0.25˙ 0.04 0.18˙ 0.01

extracted from [19, Tables 3.6 and 3.8], we conclude that the values of a, b, and c lie in the intervals .0.32, 0.58/, .0.28, 0.49/, and .0.14, 0.30/, respectively. In our calculations, we have chosen u D 0.4 m/s, z0 D 0.1 m, bu D 2.3, and two variants of bw : bw D 1.2 and bw D 1.3; the results are given in Tables 8.1 and 8.2. Here we present the results which correspond to bw D 1.2 in Table 8.1, and bw D 1.3 in Table 8.2. The tables show that in both cases the results for a and b obtained by our model agree slightly better with the experiments compared to those obtained by Thomson’s model. As to the quantity c, both models agree satisfactorily with the measurements. In conclusion we stress again that even in the case of homogeneous but anisotropic turbulence, the well-mixed condition .8.7/ does not define the 1-particle model uniquely. Here we compared our model against Thomson’s model, which gives two significantly different values of the eddy diffusivity coefficient .8.30/. To choose between these models, more accurate measurements are needed.

Chapter 9

Direct and adjoint Monte Carlo for the footprint problem Lagrangian stochastic models and algorithms are constructed and justified for solving the footprint problem, namely, the problem of calculation of the mean concentration and the flux of particles at a fixed point released from a source arbitrarily situated in space. The direct and adjoint Monte Carlo algorithms are suggested, and rigorous justifications are given. Two different backward trajectory algorithms are considered: Thomson’s method, and a method based on probabilistic representations of the relevant initial value problem. The cost of the latter algorithm may increase with time, but it allows treating the general situation when a set of reacting species is scattered by the flow.

9.1

Introduction

The footprint problem as formulated in the literature (e. g., see [55–57, 221]) essentially deals with the calculation of the contribution to the mean concentration and its flux at a fixed point from an arbitrary given source of particles. There are mainly two different approaches: (i) conventional deterministic methods based on the semiempirical turbulent diffusion equation and closure assumptions (e. g., see [221] and (ii), the stochastic approach, which utilizes trajectory simulations (e. g., see [108, 191], [198– 256]). The deterministic approach directly deals with the equation governing the mean concentration, but it is restricted by the use of the Bousinesque hypothesis, whose applicability should be additionally studied (e. g., see [19]). For instance, this hypothesis can not be true if the concentration is calculated close to the sources [19, 146]. More generally, the high-order closure methods are developed, a but different closure hypothesis also should be made [146] . Stochastic models do not require any closure hypotheses, and the main difficulty is in constructing adequate Lagrangian trajectories with the desired statistical characteristics. There are two main approaches in constructing stochastic methods. The first one is based on the Monte Carlo simulation of Eulerian random velocity fields (e. g., see [102, 191, 244]). The second approach treats the stochastic Lagrangian trajectories as solutions to the stochastic generalized Langevin equation (e. g., see [198, 215, 237]). The first approach is more rigorous, but generally requires a lot of computer time. In addition it needs detailed information about the statistical characteristics of the whole

172

Chapter 9 Direct and adjoint Monte Carlo for the footprint problem

velocity field. In contrast, the second approach needs only a 1-pont probability density function (p. d. f.) of the Eulerian velocity field, and is much more efficient in numerical calculations. It should be noted however, that this approach is rigorously justified only in the case of stationary isotropic turbulent flow. Even in the case of homogeneous but nonisotropic turbulence, the justification problem remains unsolved; in particular there are several different stochastic models which satisfy the well-mixed condition [200, 237]. In this chapter, we suggest direct and backward Monte Carlo algorithms for solving the footprint problem by simulation of Lagrangian trajectories as solutions to generalized Langevin equations. We derive random estimators for the mean concentration and its flux, both for direct and backward schemes. Two different methods are compared: the adjoint scheme, which is based on probabilistic representations of the relevant PDE [59], and the backward Thomson scheme [237]. The latter approach is extended to general stochastic differential equations.

9.2 Formulation of the problem Let us consider a passive scalar dispersed by the turbulent velocity field in the surface layer of the atmosphere. The passive scalar is assumed to follow the streamlines of the flow. We assume that the source of particles is quite arbitrary; for instance, it might be situated on the surface or in the space, or even at given points. Let us denote by q.x, t / the spatial-temporal density distribution function of the source, i. e, the number of emitted particles per unit volume in a unit time interval at the phase point .x, t /. Initially, the spatial density of particles is given by q0 .x/. The particles are transported by the 3D turbulent velocity field uE .x, t / in the surface layer D D ¹x D .x1 , x2 , x3 / : x3  0º . Let us denote by X.t ; x0 , t0 / and V.t ; x0 , t0 / the Lagrangian spatial coordinates and the velocity, respectively. The mean concentration at .x, t / is defined by [146] hc.x, t /i D c.x, N t/ (9.1) Z Z Z t dt0 d x0 q.x0 , t0 /pL .x, t ; x0 , t0 / C d x0 q0 .x0 /pL .x, t ; x0 , 0/, D 0

where

D

D

pL .x, t ; x0 , t0 / D hı.x  X.t ; x0 , t0 //i

is the probability density function (p. d. f.) of the particle’s coordinate at the time t which was started in the point x0 at the time t0 , and ı./ is the Dyrac delta-function. Here and throughout the chapter we use the notation hi for the averaging over the samples of the turbulent velocity field. We define also the concentration fluxes by Fi .x, t / D hui .x, t /c.x, t /i, where c.x, t / is the instant concentration.

i D 1, 2, 3,

173

Section 9.3 Stochastic Lagrangian algorithm

As in the case (9.1), the fluxes can be represented in the integral form (see Appendix A): Z Z t Z d uE dt0 d x0 ui q.x0 , t0 /pL .x, uE , t ; x0 , t0 / Fi .x, t / D 0 IR3 D Z Z C d uE d x0 ui q0 .x0 /pL .x, uE , t ; x0 , 0/. (9.2) IR3

D

Here u  V.t ; x0 , t0 //i pL .x, uE , t ; x0 , t0 / D hı.x  X.t ; x0 , t0 //ı.E

(9.3)

is the p. d. f. of the spatial-velocity phase point. In the analysis, it is convenient to deal with a general quantity, the spatial-velocity distribution of an ensemble of particles: Z t Z dt0 d x0 q.x0 , t0 /pL .x, uE , t ; x0 , t0 / p.x, uE , t / D D 0 Z C d x0 q0 .x0 /pL .x, uE , t ; x0 , 0/. (9.4) D

From (9.1), (9.2), and (9.4) we find Z Z p.x, uE , t /d uE , Fi .x, t / D ui p.x, uE , t /d uE , c.x, N t/ D IR3

IR3

i D 1, 2, 3.

It is of practical interest to calculate the mean concentration and relevant fluxes for arbitrarily situated surface sources. In the literature, this problem is called the footprint problem (e. g., see [55, 57, 221]. Note that in this problem the mean concentration and fluxes are evaluated at a fixed point. We consider a more general function: Z T Z Z d uE dt d x p.x, uE , t /h.x, uE , t /, .p, h/ D IR3

0

D

where h.x, uE , t / is an arbitrary function which can be chosen relevant to the quantity of interest. For instance, in the case h.y, uE , s/ D ı.yx/ı.st / we have .p, h/ D c.x, N t /. If h.y, uE , s/ D ui ı.y  x/ı.s  t /, then .p, h/ D Fi .x, t /. Thus we concentrate on the problem of calculation of the function .p, h/.

9.3

Stochastic Lagrangian algorithm

To construct algorithms based on the representations given above, we need samples of the Lagrangian trajectories X.t / D X.t ; x0 , t0 /, t  t0 . Ideally, if we had samples of the velocity uE .x, t /, the trajectories could be simulated by solving the problem d X.t / D uE .X.t /, t /, dt

t > t0

X.t0 / D x0 .

174

Chapter 9 Direct and adjoint Monte Carlo for the footprint problem

In practice one uses approximate models of the velocity field. For instance, randomized models of the Gaussian velocity fields are used (e. g., see [191]). This approach is well developed and justified only in the case of homogeneous turbulence, while the inhomogeneous case needs further development. In the general nonhomogeneous case one uses another approach based on a stochastic differential equation of the Langevin type directly governing the Lagrangian trajectory. This equation has the form [200, 215, 237] d X.t / D V.t /dt , d V.t / D a.t , X.t /, V.t //dt C

p

C0 "N.X.t /, t / d W.t /,

(9.5)

where the function a is to be specified in each specific situation, C0 is the universal Kolmogorov constant (C0  4  6), "N.x, t / is the mean dissipation rate of the kinetic energy of turbulence, and W.t / is the standard 3D Wiener process. In this section, we deal with the general scheme. Remark 9.1. Note that to complete the description of the Lagrangian stochastic model, we need to define the behavior of .X.t /, V.t // in the neighborhood of the boundary D ¹x D .x1 , x2 , x3 / : x3 D 0º. We assume that the boundary is impenetrable, i. e., that u3 .x/jx2 D 0. This implies that the Lagrangian trajectories satisfying (9.5) never do reach . Therefore it is reasonable to require that the same property holds for X.t /, the solutions to (9.5). This can be realized by special choice of the function "N.x, t / (see for details Sec. 4).

9.3.1 Direct Monte Carlo algorithm Let X.t ; x0 , t0 /, V.t ; x0 , t0 /, t  t0 be solutions to (9.5) satisfying the initial condition X.t0 / D x0 with V.t0 / D uE 0 , where uE 0 is the random velocity whose p. d. f. coincides u0 ; x0 , t0 /, the p. d. f. of the Eulerian velocity uE .x0 , t0 /. with pE .E It is convenient to use the representation .p, h/ D I C I0 , where Z Z T Z Z t Z I D d uE dt d xh.x, uE , t / dt0 d x0 q.x0 , t0 /pL .x, uE , t ; x0 , t0 / IR3 T

D and

Z dt0

0

D

0

Z

d x0 q.x0 , t0 /

Z I0 D

Z d x0 q0 .x0 /

D

Z

IR3

Z

IR3

D

d uE

dt 0

Z

T

dt t0

D

d xpL .x, uE , t ; x0 , t0 /,

Z

T

d uE

D

0

Z

D

d xh.x, uE , t /pL .x, uE , t ; x0 , t0 /.

From this representations we can write down the Monte Carlo estimators for I and I0 . Let Z Z Z T

QD

dx D

dt q.x, t /, 0

Q0 D

q0 .x/d x, D

175

Section 9.3 Stochastic Lagrangian algorithm

and let .Qx, tQ/ be a random point distributed in D Œ0, T  with the p. d. f. q.x, t /=Q, and xQ 0 be a random point distributed in D with q0 .x/=Q0 . Standard arguments of the Monte Carlo theory [191] yield Z T h.X.t ; xQ , tQ /, V.t ; xQ , tQ /, t /dt , (9.6) I D QIE.Qx,tQ/ IEW ./ tQ

eI0 D Q0 IExQ0 IEW ./ h.X.t ; xQ 0 , 0/, V.t ; xQ 0 , 0/, t /.

(9.7)

Here IE.Qx,tQ/ IEW ./ means averaging, first, over all starting points .Qx, tQ/ and, second, over all solutions of (9.5); these two averagings are taken independently. Similar notation is used in (9.7). From the probabilistic representations (9.6), (9.7) we can construct the direct Monte Carlo algorithm. For this we need a numerical scheme for solving the stochastic differential equation (9.5). For simplicity, we choose the Euler scheme. The algorithm can be described as follows. The mathematical expectation in (9.6) is usually calculated as N 1 X i , I ' N iD1

where N is the number of samples and i , i D 1, : : : , N , are independent samples of the random estimator Z DQ

tQ

T

h.X.t ; xQ , tQ/, V.t ; xQ , tQ/, t /dt .

First we choose the time step of integration as t D T =N t , N t is the total number of steps. Then the calculation of i is as follows. Step 0. S :D 0, and sample the random point .Qx, tQ/ in D Œ0, T  from the density q.x, t /=Q; u0 ; xQ , tQ/; sample uEQ 0 in IR3 from the density pE .E Q Q put t :D t ., X :D xQ , V :D uE ; Step 1. S :D S C Qt h.X, V, t /; Sample a 3D standard Gaussian random vectorp ; put X :D X C Vt , V D V C a.t , X, V/t C C0 "N.X, t /t; t :D t C t ; if t > T , then go to step 2. Otherwise go back to start the step 1. Step 2. put i :D S ;

176

Chapter 9 Direct and adjoint Monte Carlo for the footprint problem

The calculation of I0 is treated analogously: I0 '

N 1 X i N iD1

where N is the number of samples, i , i D 1, : : : , N are independent samples of the random estimator  D Q0 h.X.t ; xQ 0 , 0/, V.t ; xQ 0 , 0/, t /. The calculation of i : Step 0. Sample a random point xQ 0 in D from the density q0 .x/=Q0 ; u0 ; xQ 0 , 0/; sample uEQ 0 in IR3 from the density pE .E put X :D xQ 0 , V :D uEQ 0 ; t :D 0. Step 1. Sample a 3D standard Gaussian random vector ; p N /, t /t ; t :D t C t ; put X :D X C Vt , V D V C a.t , X, V/t C C0 ".X.t if t  T , then go to step 2. Otherwise go to start the step 1. Step 2. put i :D Q0 h.t , X, V/;

9.3.2 Adjoint algorithm Clearly, the direct algorithm is not practically applicable in the general situation if the function h.x, uE , t / is concentrated on small domains. In this case, an adjoint scheme is preferable. For this purpose we need the probabilistic representation for p.x, uE , t /. In what follows we use the summation convention. The function pL .x, uE , t ; x0 , t0 / defined in (9.3) satisfies the Kolmogorov–Fokker–Planck equation for the stochastic differential equation (9.5): 1 @2 @.ui pL / @.ai pL / @pL C D C0 ".x, N t/ pL .x, uE , t ; x0 , t0 / C @t @xi @ui 2 @ui @ui with the initial conditions

ˇ pL .x, uE , t ; x0 , t0 /ˇ tDt0 D ı.x  x0 /pE .E u; x0 , t0 /.

From this, using (9.4) we find @p 1 @2 p @.ui p/ @.ai p/ C D C0 ".x, N t/ p.x, uE , t / C q.x, t /pE .E u; x, t / C @t @xi @ui 2 @ui @ui and

ˇ p.x, uE , t /ˇ tD0 D q0 .x/pE .E u; x, 0/.

177

Section 9.3 Stochastic Lagrangian algorithm

Using the probabilistic representation presented in Appendix B we get ´Z t E E E E O x,u,t O x,u,t O x,u,t O x,u,t p.x, uE , t / D IE.x,u,t/ q.X E t0 , V t0 , t0 /pE .V t0 ; X t0 , t0 / 0

 Z t E E E 0 0 O x,0u,t O x,0u,t O x,u,t exp  .X , V , t /dt / t0 t0 0 0 dt0 C q0 .X0 t0



u,t E u,t E O x, O x, ;X , 0/ exp pE .V 0 0

(9.8)

 Z t μ x,u,t E x,u,t E 0 0 O t0 , V O t 0 , t0 /dt0 ,  .X 0 0 0

E E .x,u,t/ E O 0/  X O x,u,t O O x,u,t , and X.t where .x, uE , t / D @ai@u t0 , V.t0 /  V t0 , t0  t is the adjoint i trajectory determined as the solution to the equation

O 0 /dt0 , O 0 / D V.t d X.t O 0 / D a.t0 , X.t O 0 /, V.t O 0 //dt0 C d V.t with the initial conditions

ˇ O 0 /ˇˇ X.t

t0 Dt

D x,

q

O 0 /, t0 / d W.t O 0 /, C0 "N.X.t

ˇ O 0 /ˇˇ V.t

t0 Dt

D uE .

O 0 / is a standard 3D Wiener Note that here a./ is the same function as in (9.5), and W.t process. From the probabilistic representation (9.8) we get Z T Z Z d uE dt d x h.x, uE , t /IE.x,u,t/ .p, h/ D E 0 IR3 D ´Z t E E E E O x,u,t O x,u,t O x,u,t O x,u,t q.X ,V , t0 /.pE .V ;X , t0 / 0

t0

t0

t0

t0

 Z t u,t E x,u,t E 0 0 O x, O 0 0 exp  .X , V , t /dt t0 t0 0 0 dt0 t0

C

u,t E u,t E u,t E O x, O x, O x, q0 .X / pE .V ;X , 0/ exp 0 0 0

 Z t μ x,u,t E x,u,t E O t0 , V O t 0 , t00 /dt00 .  .X 0 0 0

Now we are in position to write the Monte Carlo estimator for IO and IO0 . Let .x, uE , t / be an arbitrary probability density function in D IR3 Œ0, T  which satisfies the condition .x, uE , t / ¤ 0 if h.x, uE , t / ¤ 0. Let .Ox, uEO , tO/ be a random point distributed in D IR3 Œ0, T  with density . Then O .p, h/ D IE.Ox,u, IE O , EO tO/ W./

178

Chapter 9 Direct and adjoint Monte Carlo for the footprint problem

where " ´  Z tO μ O O Z tO 0 0 O D h.Ox, uE , t / O O O O O .X t00 , V t00 , t0 /dt0 dt0 q.X t0 , t0 /pE .V t0 ; X t0 , t0 / exp  t0 .Ox, uEO , tO/ 0  Z tO # 0 0 O O O O O C q0 .X0 /pE .V0 ; X0 , 0/ exp  .X t00 , V t00 , t0 /dt0 . 0

Here we used a brief notation O O

Et O xtO ,u, O t0  X , X 0

O O

Et O t0  V O xtO ,u, V . 0

The calculations are carried out according to .p, h/ '

N 1 XO i , N iD1

where N is the number of samples, Oi , i D 1, : : : , N , are independent samples of the O random estimator . The algorithm of calculation of the sample Oi : Step 0. S :D 0; sample the random point .Ox, uEO , tO/ in D IR3 Œ0, T  from the density .x, uE , t /; EO tO/ ; put t :D tO, X :D xO , V :D uEO ; :D 1. Q :D h.Ox,u, .Ox,u, EO tO/

Step 1. S :D S C t q.X, t /pE .V; X, t / ; Sample a 3D standard Gaussian random vectorp; N t /t ; put X :D X  Vt , V D V  a.t , X, V/t C C0 ".X,

:D exp..X, V, t /t /; t :D t  t ; if t < 0, then go to step 2. Otherwise go back to start the step 1. Step 2. put Oi :D QŒS C q0 .X/pE .V; X, 0/ .

9.4 Impenetrable boundary Note that to complete the description of the Lagrangian stochastic model we need to define the behavior of X.t /, V.t /, the solution to (9.5) in the neighborhood of the boundary D ¹x : z D x3 D 0º. We assume that the boundary is impenetrable,

179

Section 9.4 Impenetrable boundary

i. e., that w D u3 D 0 at the boundary of . This implies that the true Lagrangian trajectories never reach . Therefore it is reasonable to require that the same property holds for X.t /, the solutions to (9.5). This can be done by the special choice of the function ".z, N t /. Indeed, in the neighborhood of it is reasonable to consider the flow as neutrally stratified. Therefore, pE .w/ is Gaussian, with constant w , and the vertical profile of ".z/ N is given by [146] ".z/ N D

u3 , z

' 0.4.

(9.9)

Here is the Karman constant, and z0 is the roughness height. The equation of vertical motion Z.t / D X3 .t /, W .t / D V3 .t / is then dZ D W dt , where aD

d W .t / D  C0 u3 , 2 w2

b a W .t /dt C p dB.t /, Z Z

bD

(9.10)

 C u3  12 0  .

If we assume that the formula (9.9) is true for all z > 0, then all the solutions to (9.10) do not reach the boundary . Indeed, let  be a random time (which depends on the trajectory Z.t /) defined by Z  .t / D 0

t

ds . Z.s/

Then, the vertical velocity in the new time variable W . / satisfies the equation d W . / D aW . /d  C b dB. /. Therefore, from

dZ dt dZ D D W . /Z. / d dt d 

we have

Z Z. / D Z.0/ D exp¹S. /º,

S. / D



W . 0 / d  0 .

0

The function W . / is an Uhlenbeck–Ornstein process with continuous samples. Therefore, jS. /j < 1 with probability 1 for arbitrary  > 0. This implies that Z. / > 0 provided that Z.0/ > 0. Thus the function Z. / never reaches the boundary . The same is true for Z.t /. To show this, it is sufficient to note that t . / ! 1 as  ! 1. Let us show this property. We have Z Z Z dt 0 0 0 d D Z. / d  D Z.0/ exp¹S. 0 /º d  0 . t . / D 0 0 d 0 0

180

Chapter 9 Direct and adjoint Monte Carlo for the footprint problem

In [108] it is shown that with probablity 1 Z 1 exp¹S. /º d  D 1. 0

This implies that with probability 1 t . / ! 1 as  ! 1. In numerical implementation, it is convenient to simulate the trajectory in the neighborhood of by numerical solution to d ln Z. / D W . /, d d W . / D aW . / C b dB. /, dt . / D Z. /. d The Euler scheme reads Z. C  / D Z. / exp¹W . / º,

p W . C  / D W . /  aW . / C b  , t . C  / D t . / C Z. / , where  is the discretization step, and  is a standard normal random number. Note that by construction, this scheme ensures that the boundary is impenetrable, i. e., Z. / > 0 for all  .

9.5 Reacting species In this section we show that the backward trajectory technique can be extanded to the case where a set of reacting species is in play. Assume that in the domain D with a boundary which is impenetrable there are K reacting species governed in Lagrangian formulation by the system d X.t / D V.t /dt ,

p d V.t / D a.t , X.t /, V.t //dt C C0 "N.X.t /, t / d W.t /, dNk D fk .t , X, N/, k D 1, 2. : : : , K. dt with initial conditions X.0; x0 / D x0 ,

V.0; x0 / D V0 ,

(9.11)

Nk .0; x0 / D qk .x0 /, k D 1, : : : , K,

where V0 is the initial random velocity whose p. d. f. is pE .V; x0 /, and qk .x0 / is the initial spatial distribution of the species, N D .N1 , : : : , NK /. The Eulerian concentration of k-th specy is given by Z d x0 Nk .t ; x0 /ı.x  X.t ; x0 //, k D 1, : : : , K. nk .x, t / D D

181

Section 9.5 Reacting species

Then the mean is hnk .x, t /i D

Z

Z dV

Z dN

IR3

d x0 Nk pL .x, V, N, t ; x0 /, D

k D 1, : : : , K,

where pL .x, V, N, t ; x0 / D hı.x  X.t ; x0 //ı.V  V.t ; x0 //ı.N  N.t ; x0 //i. This p. d. f. satisfies the Kolmogorov–Fokker–Planck equation @ @ C0 "N @2 pL @ @pL .Vi pL / C .ai pL / C .fk pL / D C @t @xi @Vi @Nk 2 @Vi @Vj with the initial condition pL .x, V, N, 0; x0 / D ı.x  x0 / pE .V; x0 /ı.N  q.x0 //. Here q.x0 / D .q1 .x0 /, : : : , qK .x0 //. The flux of k-th specy in the i -th direction is given by Z Z Z dV dN d x0 Vi Nk pL .x, V, N, t ; x0 /, k D 1, : : : , K. hui .x, t /nk .x, t /i D IR3

D

Generally, the integral Z Z Z dV dN d x0 h.x, V, N, t /pL .x, V, N, t ; x0 /, Ih .x, t / D IR3

D

k D 1, : : : , K (9.12)

is to be evaluated for a given function h. Let Z d x0 pL .x, V, N, t ; x0 /. .x, V, N, t / D D

Then this function satisfies @ @ C0 "N @2  @ @ .Vi  / C .ai  / C .fk  / D C @t @xi @Vi @Nk 2 @Vi @Vj and the initial condition .x, V, N, 0/ D pE .V; x/ı.N  q.x//. The probabilistic representation given in Appendix B in this case yields ² Z t ³ @ai x,V,t x,V,t x,V,t x,V,t .s, Xs , Vs /ds (9.13) .x, V, N, t / D IEpE .V0 ; X0 / exp  0 @Vi ³ ² Z t   @fk x,V,N,t .s, Xx,V,t , N /ds ı Nx,V,N,t  q.Xx,V,t / , exp  s s 0 0 0 @Nk

182

Chapter 9 Direct and adjoint Monte Carlo for the footprint problem

where X.s/ D Xx,V,t , V.s/ D Vx,V,t and N.s/ D Nx,V,N,t , 0  s  t are the adjoint s s s Lagrangian processes defined as the solutions to d X.s/ D V.s/ds,

p d V.s/ D a.s, X.s/, V.s//ds C C0 "N.X.s/, s/ d W.s/, dNk D fk .s, X, N/, k D 1, 2. : : : , K, ds with the terminal conditions X.t / D x,

V.t / D V,

Nk .t / D Nk , k D 1, : : : , K.

The expectation IE is taken over all backward Lagrangian trajectories. From (9.12) and (9.13) we find Z Z d V d N h.x, V, N, t / Ih .x, t / D IR3

where

; Xx,V,t /Qa .x, V, t /Qf IEpE .Vx,V,t 0 0 ² Z Qa .x, V, t / D exp 

t

0

² Z Qf .x, V, N, t / D exp 

0

Hence

t

.x, V, N, t /ı.Nx,V,N,t  q.Xx,V,t //, 0 0

³ @ai x,V,t .s, Xx,V,t , V /ds , s s @Vi

³ @fk x,V,t x,V,N,t .s, Xs , Ns /ds . @Nk

Z

Ih .x, t / D IE

d VpE .Vx,V,t ; Xx,V,t /Qa .x, V, t / 0 0

IR3

Z



 q.Xx,V,t //. d N Qf .x, V, N, t /h.x, V, N, t /ı.Nx,V,N,t 0 0

(9.14)

To evaluate the last integral in (9.14) we use a known formula. Let F be a function of x D .x1 , : : : , xm /, and g be a vector function g.x/ D .g1 .x/, : : : , gm .x//, whose inverse g 1 exists, and let b D .b1 , : : : , bm / be a fixed vector. Then Z F .xb / , (9.15) F .x/ı.g.x/  b/ d x D J.xb / where xb D g 1 .b/, and

ˇˇ ˇˇ ˇˇ @gi .x/ ˇˇ ˇˇ ˇ ˇ J.x/ D Det ˇˇ @xj ˇˇ

is the Jacobian of the transformation x ! g.x/. Applying (9.15), we can evaluate the has the Jacobian last integral in (9.14). Indeed, the transformation g : N ! Nx,V,N,t 0

183

Section 9.6 Numerical simulations

Qf .x, V, N, t /. Choosing b D q.Xx,V,t / we find that g 1 .b/ D N.t ; b/ where N.s; b/ 0 is the solution to dN , s/, N.0; b/ D b. D f .N, Xx,V,t s ds Thus we have Z d VpE .Vx,V,t ; Xx,V,t /Qa .x, V, t /h.x, V, N.t ; q.Xx,V,t //, t /. Ih .x, t / D IE 0 0 0 IR3

Now we can formulate the backward algorithm for solving problem (9.11). Introduce Q for a sample a probability density function r.V/ > 0 in IR3 , and use the notation V Q Q Q t0 D Vx,V,t Q t0 D Xx,V,t and V from this density. We write for simplicity X t0 t0 . We denote Q by N.s/ the solution to Q dN Q Q 0 /. Q X Q s , s/, N.0/ D q.X D f .N, ds The random estimator has the form: Q t / D 1 pE .V Q 0; X Q 0 / Qa .x, V, Q t /h.x, V, Q N.t Q /, t /, .x, V, Q r.V/ and hence

Q t /. Ih .x, t / D IEVQ IE.x, V,

Q and IE is the averaging over the trajecHere IEVQ means averaging over the samples V, Q s, V Q s, 0  s  t. tories X

9.6

Numerical simulations

Numerical simulations are carried out for the following problem. A horizontally homogeneous stationary neutrally-stratified surface layer is considered. The source starts to generate particles at time t D 0. It is uniformly distributed over the plane at height z D zs . In this section we use the SI metric system of units. We calculate the mean concentration and its vertical flux at a fixed point at height z D zd by three methods: the direct Monte Carlo described in Section 9.3.1, and two backward in time Monte Carlo algorithms presented in Section 9.3.2 and in Appendix C: the adjoint algorithm based on the probabilistic representation, and the backward method due to Thomson [237]. Comparison of the cost of these three methods is given. Since our problem is described by horizontally homogeneous parameters, it is governed by a 1-dimensional stochastic model (9.9), (9.10). First, let us present the details of random estimators for the direct estimator of the type (9.6). In our case it takes the form c.z N d , t / D Q IE

 tX .zd / j D1

1 jW .j /j

(9.16)

184

Chapter 9 Direct and adjoint Monte Carlo for the footprint problem

for the mean concentration and F .zd , t / D Q IE

 tX .zd / j D1

W .j / jW .j /j

(9.17)

for the vertical concentration flux. Here .Z. /, W . // is the solution to (9.10) with the initial condition Z.0/ D zs , W .0/ D w0 , where w0 is a random p velocity distributed 2 2 with the Eulerian Gaussian p. d. f. pE .w/ D exp¹w =2w º= 2w2 . The values j are random times at which the process Z.s/ intersects the level z D zd , and  t .zd / is the total number of such events in the interval 0  s  t . The constant Q is the strength of the surface area source. The derivation of the estimator (9.16) can be obtained as follows. Since Z 1 Z t Z 1 dw dt0 dz0 ı.z0  zs / c.z N d , t/ D 1 0 0 Z 1 dw0 pE .w0 /pL .zd , w, t ; z0 , w0 , t0 / Z D

1 1

Z

dw 1

Z



1

0 Z t

D IE 0

Z

t

dt0 0

1

Z dz0 ı.z0  zs /

0

1 1

dw0 pE .w0 /

dz0 ı.z  zd /pL .z, w,  ; z0 , w0 , 0/

ı.Z zs ,wQ 0  zd / d  .

(9.18)

Here wQ 0 is a random number distributed with pE .w0 /, Z tz,w is the solution to the system (9.10) with the initial deterministic condition Z0z,w D z, and W0z,w D w. It remains to note that for arbitrary continuously differentiable function Z. /, Z

t

ı.Z. /  z/ d  D 0

X t .z/ j D1

1 ˇ ˇ, ˇ dZ. j / ˇ ˇ d ˇ

where  t .z/ is the number of intersections of the level z by the trajectory Z. / in the interval 0    t , and j are the intersection times. The random estimator of Thomson’s backward algorithm used in [57] is constructed on the solutions to the backward-time stochastic differential equation O 0 / D WO .t0 /dt0 , d Z.t

a b d WO .t0 / D WO .t0 /dt0 C p dB.t0 /, ZO ZO

185

Section 9.6 Numerical simulations

O / D zd , WO .t / D w, with terminal condition Z.t O where wO is a random number distributed with Gaussian pE .w/ given above. Here aD

C0 u3 , 2 w2

bD

 C u3  12 0  .

The random estimators read c.z N d , t / D Q IE

X t .zs / j D1

1 O jW .j /j

(9.19)

for the mean concentration and F .zd , t / D Q IE

X t .zs / j D1

WO .t / jWO .j /j

(9.20)

for the vertical concentration flux. O 0 / (0  t0  t ) Here the values j are random times at which the process Z.t intersects the level z D zs , and  t .zs / is the total number of such events in the interval 0  t0  t . Finally, let us consider the adjoint estimator based on probabilistic representation. The probabilistic representation (9.8) in our case (1D) reads ²Z t ³ Z t b ds Q Q ı.Z.t0 /  zs /pE .W .t0 // exp (9.21) dt0 , p.z, w, t / D IEz,w Q 0 t0 Z.s/ Q 0 / D Z tz,w , WQ .t0 // D W tz,w , 0  t0  t is the adjoint trajectory defined by where Z.t 0 0 Q 0 / D WQ .t0 / dt , d Z.t

a b d WQ .t0 / D  WQ .t0 /dt0 C p dB.t0 /, Q Z ZQ

Q / D z, WQ .t / D w. with terminal condition Z.t Now, substituting (9.21) in Z c.z N d , t/ D

C1

1

p.zd , w, t / dw

we get Z c.z N d , t/ D

C1

1

dwIEzd ,w

QX t .zs / j D1

pE .WQ .j // exp jWQ .j /j

²Z

t

j

³ b ds . Q Z.s/

(9.22)

186

Chapter 9 Direct and adjoint Monte Carlo for the footprint problem

Here we used the relation ²Z Z t Q Q ı.Z.t0 /  zs /pE .W .t0 // exp 0

t t0

³ b ds dt0 Q Z.s/ D

QX t .zs / j D1

pE .WQ .j // exp jWQ .j /j

²Z

t

j

³ b ds . Q Z.s/

where Q t.zs / is the number of intersections of the level z D zs by the trajectory Z tz0d ,w , 0  to  t , and 1 , : : : , j , : : : are the intersection times. The resulting estimator is ´ ²Z t ³μ Q t .zs / pE .W zjd ,wQ / 1 X b ds . (9.23) exp c.z N d , t / D IE z ,w Q r.w/ Q j Z d jW zd ,wQ j j D1

j

where r.w/ is a p. d. f. which is positive in .1, C1/, and wQ is a random number distributed with r.w/. Analogous arguments lead to the random estimator for the vertical concentration flux: ´ ²Z t ³μ Q t .zs / pE .W zjd ,wQ / wQ X b ds F .zd , t / D IE exp . (9.24) z ,w Q r.w/ Q j Z d jW zjd ,wQ j j D1 In numerical calculations, one takes a cut-off in the integral over w in (9.22), and integrates from, say, A to A, A being sufficiently large; in our case we have chosen A D 5w . Further parameters in calculations are w D 1.25 u , u D 0.4, and the function "N is defined in (9.9). As to the penetrable boundary conditions, in calculations it cannot be strictly satisfied. In the numerical scheme it is convenient to follow the trajectories until some reflection height z D zr < z0 , and then reflect them according to some reflection law. In calculations we found that beginning from zr < zs =5, the results are stable with respect to further decreasing of the reflection height zr . The perfect reflection (symmetric to the plane z D zr ) was used. It should be noted that usually (e. g., see [55]) one reflects the trajectories at the height z D z0 , which does not affect the calculations at large (compared to z0 ) heights, but at a height of several z0 the error may be about 10–30 %. In Table 9.1 we present the mean concentration and its vertical flux obtained by the direct (9.16), (9.17), the adjoint (9.23), (9.24), and the backward method (9.19), (9.20). The calculations are made for four time instances t D TL .zs /, 2TL .zs /, 4TL .zs /, and 8TL .zs /, for zd D 1, zs D 0.5. The unit source strength uniformly distributed over the plane z D zs was taken. The N s/ D Lagrangian time scale TL D TL .zs / is given by [215] TL .zs / D 2w2 =C0 ".z zs =a D 0.39. The error of the results presented in the table was less than 1 % calculated

187

Section 9.7 Conclusion Table 9.1. Comparison of three different methods. Method

t D TL

t D 2TL

t D 4TL

t D 8TL

direct

cN F cost

2.08e  3 2.71e  3 200 [s]

8.94e  2 6.48e  2 200 [s]

4.67e  1 2.13e  1 85 [s]

1.26 0.4 60 [s]

adjoint

cN F cost

1.95e  3 2.51e  3 400 [s]

8.42e  2 6.10e  2 1,800 [s]

4.71e  1 2.16e  1 2,000 [s]

1.19 0.34 2,200 [s]

backw.

cN F cost

1.95e  3 2.50e  3 180 [s]

9.03e  2 6.58e  2 300 [s]

4.68e  1 2.05e  1 300 [s]

1.21 0.37 40 [s]

p as the statistical error measured as 3 standard deviation= number of samples. The cost means the computer time of a 233 MHz PC computer. The results presented in the table show that in this special case of horizontally homogeneous problem the direct and backward methods have approximately equal cost and are both much more efficient than the adjoint method. It should be stressed, however, that this is because our model problem is actually 1-dimensional and the source is stationary. In the general case of 3D problems with a source generating particles during a short period of time, the backward algorithm is much more efficient. As to the adjoint method, although our calculations show that it requires a lot of computer time, it has the following important advantages. The method allows us to solve problems of transport of reacting species as described in Section 9.5. Another advantage of the adjoint method is in treating the problems with boundary conditions. It should be noted that the boundary value problems cause difficulties for the backward method.

9.7

Conclusion

A generalized footprint problem is treated as a calculation of a linear integral over space, velocity and time of the space-velocity distribution of ensemble of particles in a turbulent flow. The Lagrangian stochastic description is used to solve this problem. As important particular cases the mean concentration and its flux are analysed in details. Three different algorithms are presented: (1) direct Monte Carlo, (2) adjoint Monte Carlo, and (3) backward Monte Carlo algorithms. The direct Monte Carlo is quite general but it is not efficient in estimation of local functions like, e.g, the concentration and its flux at a fixed point. The adjoint method is also general and is especially convenient for evaluation of local functionals. The method is based on the well developed probabilistic representations

188

Chapter 9 Direct and adjoint Monte Carlo for the footprint problem

for the boundary value problems. Therefore, it allows to solve problems with quite general boundary conditions. Unfortunately, the method requires a lot of computer time because the variance increases with time instance under consideration. The backward algorithm originally presented by Thomson is extended to more general case when the transport in the phase space is described by general stochastic differential equation. This extension allows to treat problems with absorption, which is of our current interest.

9.8 Appendices 9.8.1 Appendix A. Flux representation Here we derive the representation (9.2). By the definition, the instant concentration is Z Z t Z d x0 q.x0 , t0 /ı.x  X.t ; x0 , t0 // C d x0 q0 .x0 /ı.x  X.t ; x0 , 0//. c.x, t / D dt0 0

D

D

From this, Fi .x, t / D hui .x, t /c.x, t /i Z t Z D dt0 d x0 q.x0 , t0 /hui .x, t /ı.x  X.t ; x0 , t0 //i 0Z D d x0 q0 .x0 /hui .x, t /ı.x  X.t ; x0 , 0//i. C

(A1)

D

Since ui .x, t /ı.x  X.t ; x0 , t0 // D ui .X.t ; x0 , t0 /, t /ı.x  X.t ; x0 , t0 // D Vi .t ; x0 , t0 /ı.x  X.t ; x0 , t0 //, then hui .x, t /ı.x  X.t ; x0 , t0 //i D hVi .t ; x0 , t0 /ı.x  X.t ; x0 , t0 //i Z Z dV d X Vi ı.x  X/pL .X, V, t ; x0 , t0 / D IR3 D Z D Vi pL .x, V, t ; x0 , t0 / d V. IR3

From this we get in view of (A1) the desired representation (9.2).

9.8.2 Appendix B. Probabilistic representation Let us write down the probabilistic representation of the solution .y, t / D .y1 , : : : , yn , t /, to the following general parabolic equation: @ 1 @2  @ C Bij .y, t / C ˇ.y, t / C f .y, t / D 0, C Ai .y, t / @t @yi 2 @yi @yj

t 2 Œ0, T /,

189

Section 9.8 Appendices

with the terminal condition .y, t /j tDT D f0 .y/. The probabilistic representation to this problem has the form [59]: ´Z .y, t / D IE.y,t/

T

Z

y,t

s

f .Ys , s/ exp t

y,t ˇ.Y ,  /d  ds

t

C

y,t f0 .YT / exp

Z

T

y,t ˇ.Y ,  /d 

μ ,

(B1)

t

y,t

where Ys  Y.s/, s  t solves the problem d Yi .s/ D Ai .Y.s/, s/ds C ij .Y.s/, s/d Wj .s/,

Y.s/jsDt D y. (B2)

s > t,

Here ik j k D Bij . In (B1), IE.y,t/ stands for the expectation taken over all trajectories (solutions to (B2)) starting from y at time t . Note that the solution to the equation @ 1 @2  @ C ˇ.y, t / D Bij .y, t / C f .y, t /, C Ai .y, t / @t @yi 2 @yi @yj

t 2 Œ0, T /,

satisfying .y, 0/ D f0 .y/, has the probabilistic representation ´Z

t

.y, t / D IE.y,t/ 0

 y,t f .X t0 , t0 / exp

Z  C

t t0

y,t ˇ.X t 0 , t00 /dt00 0





y,t f0 .X0 / exp

dt0 Z

 0

t

μ y,t ˇ.X t0 , t0 /dt0

,

y,t

where X t0  X.t0 /, 0  t0  t solves the problem dXi .t0 / D Ai .X.t0 /, t0 /dt0 C ij .X.t0 /, t0 /d Wj .t0 /, 0  t0  t ,

9.8.3

X.t0 /j t0 Dt D y.

Appendix C. Forward and backward trajectory estimators

In this Appendix we treat the evaluation of the integral: Z Ih,q D

Z dy

D

Z

T

dt 0

D

Z d y0

0

t

dt0 h.y, t /q.y0 , t0 /p f .y, t ; y0 , t0 /,

(C1)

190

Chapter 9 Direct and adjoint Monte Carlo for the footprint problem

where D is a domain in IRn , T > 0, h and q are functions defined in D Œ0, T , y ,t and p f .y, t ; y0 , t0 / D hı.y  Y t 0 0 /i is the transition density of the n-dimensional y0 ,t0 diffusion process Y t , the solution to d Yi .t / D Ai .Y.t /, t /dt C ij .Y.t /, t /d Wj .t /,

s > t,

Y.t /j tDt0 D y0 . (C2)

We assume that the boundary of D is penetrable, i. e., the trajectories determined by (C2) do not reach the boundary. The direct Monte Carlo estimator for evaluating the integral (C1) is straightforward: Z Z T Z T Z d y0 dt0 dy dt h.y, t /q.y0 , t0 /p f .y, t ; y0 , t0 / Ih,q D D

0

D

t0

³ ² Z T q.Qy0 , tQ0 / yQ ,tQ h.Y t 0 0 , t / dt . D IE p0 .Qy0 , tQ0 / tQ0 Here p0 .y0 , t0 / is an arbitrary p. d. f. in D Œ0, T  consistent with the function q.y0 , t0 / in the sense that p0 .y0 , t0 / > 0 if q0 .y0 , t0 / ¤ 0, and the expectation is taken over all yQ ,tQ sample points .Qy0 , tQ0 / and sample trajectory Y t 0 0 , tQ0  t  T ; the random points yQ 0 , tQ0 are distributed with p0 .y0 , t0 /. A backward estimator can be obtained by a technique suggested by Thomson [237]. Assume that we have a positive function .y, t / defined on D Œ0, T  as a solution to the equation 1 @2 .Bij / @ @ .Ai / D , .C3/ C @t @yi 2 @yi @yj y,t

where ik j k D Bij . Let p b .y0 , t0 ; y, t / D hı.y0  Z t0 /i is the transition density of the diffusion process

y,t Z t0 ,

0  t0  t which is defined by

dZi D Ai .Z, t0 / dt0 C Bij .Z, t0 /, t0 / d Wj .t0 /,

t0 < t ,

Z.t / D y.

.C4/

Here Ai .y, t / D Ai .y, t / 

1 @ .Bij .y, t /.y, t //. .y, t / @yj

We again assume that the solutions to (C4) never reach the boundary of D. Then the following relation is true: .y0 , t0 /p f .y, t ; y0 , t0 / D .y, t /p b .y0 , t0 ; y, t /. To prove this, we first remark that the function p b and F .y0 , t0 ; y, t / D

.y0 , t0 / f p .y, t ; y0 , t0 / .y, t /

.C5/

191

Section 9.8 Appendices

satisfy the equations Ly0 ,t0 F D 0,

Ly0 ,t0 p b D 0,

where the operator Ly0 ,t0 acts on a function g.y0 , t0 / as follows: Ly0 ,t0 g D

@ @g 1 @2 .Bij g/ C .Ai g/ C . @t @y0i 2 @y0i @y0j

Since the values of the functions F and p b at t0 D t coincide, p b .y0 , t ; y, t / D F .y0 , t ; y, t / D ı.y  y0 /,

.C6/

we conclude that F  p b , provided that the equation Ly0 ,t0 g D 0 with the initial condition g.t , y0 / D ı.y0  y/ has a unique solution. This implies that (C5) is true. Now we note that Ly0 ,t0 p b D 0 is true indeed, since it is the first Kolmogorov equation for p b . The equality Ly0 ,t0 F D 0 then follows from (C3), the inverse Kolmogorov equation @p f 1 @2 p f @p f .y, t ; y0 , t0 / C Ai .y0 , t0 / C Bij .y0 , t0 / D0 @t0 @y0i 2 @y0i @y0j and the expression for Ai .y0 , t0 / through Ai and Bij given above. Now, we present the backward Monte Carlo algorithm based on the (C5). We proceed as follows. Substituting the expression for p f from (C5) into (C1) we get Z Z T Z Z t .y, t / b Ih,q D dy dt d y0 dt0 h.y, t /q.y0 , t0 / p .y0 , t0 ; y, t /. .C7/ .y0 , t0 / 0 0 D D Let r.y, t / be a probability density in D Œ0, T  consistent with h, i. e., r > 0 if h ¤ 0. Then from (C7) we get ´ μ Z yQ ,tQ h.Qy, tQ/.Qy, tQ/ tQ q.Z t0 , t0 / dt0 . .C8/ Ih,q D IE Q r.Qy, tQ/ 0 .ZyQ ,t , t / t0

0

Here the expectation is taken over the random points .Qy, tQ/ distributed in D Œ0, T  yQ ,tQ with density r.y, t /, and backward trajectories Z t0 , 0  t0  tQ. Another backward trajectory estimator which generalizes the estimator presented in Section 3.2 can be obtained as follows. Let Z t Z d y0 dt0 q.y0 , t0 /p f .y, t ; y0 , t0 /. .y, t / D D

0

This function solves the problem 1 @2 .Bij  / @.Ai  / @ D C q.y, t /, C @t @yi 2 @yi @yj

.y, 0/ D 0.

.C9/

192

Chapter 9 Direct and adjoint Monte Carlo for the footprint problem

From the probabilistic representation given in Appendix B we get ´Z ² Z t ³ μ t y,t y,t .y, t / D IE q.X t0 , t0 / exp  R.Xs , s/ ds dt0 , t0

0

y,t

where the expectation is taken over the backward trajectories X t0 , 0  t0  t determined from dXi .t0 / D AOi .X.t0 /, t0 / dt0 C Cij .X.t0 /, t0 / d Wj .t0 /, t0  t , X.t / D y. Here

@Bik .x, t / AOi .x, t / D Ai .x, t / C 2 @xk

and R.x, t / D Thus we have Z Z dy Ih,q D D

Z dy

D

T

dt h.y, t /.y, t /

0

Z D

@Ai .x, t / @2 Bij .x, t / C . @xi @xi @xj

²Z

T

dt h.y, t /IEX 0

0

t

° Z t ± ³ y,t y,t q.X t0 , t0 / exp  R.Xs , s/ ds dt0 . t0

y,t X t0 ,

0  t0  t . From this we find Here IEX stands for the averaging over trajectories ´Z ² Z tQ ³ μ! tQ h.Qy, tQ/ yQ ,tQ yQ ,tQ Ih,q D IE .C10/ q.X t0 , t0 / exp  R.Xs , s/ ds dt0 , r.Qy, tQ/ 0 t0 where yQ , tQ is a random point distributed in D Œ0, T  with density r.y, t / consistent with h.y, t /. The notation IE means the expectation over the random points yQ , tQ and yQ ,tQ trajectories X t0 , 0  t0  tQ. Remark 9.2. In this appendix we assumed that the boundary is impenetrable. However, in practice one also treats situations where a part of boundary (say, the upper bound of a layer) can be reached by the Lagrangian trajectories. In this case the boundary conditions should be given. For instance, an absorption, reflection or other behavior at the boundary can be considered. For the direct algorithm this can be taken into account by simulating the relevant behavior of the trajectories of (C2) at the boundary (e. g., the trajectories are absorbing at the absorbing boundary, reflecting at the reflection boundary, etc.). In the adjoint algorithm the situation is more complicated. Indeed, it is not clear how to arrange the behavior of trajectories, the solutions to (C4), to guarantee that (C5) is fulfilled. Note that in the approach based on (C10) there is a need for the generalization of probabilistic representation of the solution to (C9) with the relevant boundary conditions. Such representation can be derived from the known probabilistic representations [59].

Chapter 10

Lagrangian stochastic models for turbulent dispersion in an atmospheric boundary layer A 1-particle 3-dimensional stochastic Lagrangian model for transport of particles in a horizontally homogeneous atmospheric surface layer with arbitrary 1-point probability density function of Eulerian velocity fluctuations is suggested. A uniquely-defined Lagrangian stochastic model in the class of well-mixed models is constructed from physically plausible assumptions. These assumptions are (i) in the neutrally-stratified horizontally homogeneous surface layer, the vertical motion is mainly controlled by eddies whose size is of order of the current height, and (ii) the streamwise drift term is independent of the crosswind velocity. Numerical simulations for neutral stratification have shown a good agreement of our model with the well-known Thomson model, with Flesch and Wilson’s model, and with experimental measurements as well. However there is a discrepancy in these results to the results obtained by Reynolds’ model.

10.1

Introduction

This chapter deals with 1-particle Lagrangian stochastic models for 2-dimensional (2D) and 3-dimensional (3D) turbulent transport. Here we treat the flow in the atmospheric boundary layer as a fully developed turbulence (i. e., a flow with very high Reynolds number) and consider it to be a random velocity field .u, v, w/ assumed to be incompressible. Therefore, the trajectories of particles in such flows are stochastic processes. To simulate these stochastic processes, two different approaches are known in the literature. The first one is based on the numerical solution of the system of random equations @X D u.X , Y , Z, t /, @t

@Y D v.X , Y , Z, t /, @t

@Z D w.X , Y , Z, t /. @t

(10.1)

Here X.t /, Y .t /, Z.t / are the coordinates of the Lagrangian trajectory at the time t . The random fields u, v, w are simulated by Monte Carlo methods (e. g., see [42, 63, 102, 191, 198, 244], and the random trajectories are then obtained by the numerical solution of .10.1/ with the relevant initial data. In the second approach the true trajectory X.t /, Y .t /, Z.t / is assumed to be approxO /, a solution to a stochastic differential imated by a model trajectory XO .t /, YO .t /, Z.t

194

Chapter 10

Dispersion in atmospheric boundary layer

equation of Ito type (e. g., see [215,237,256] and the list of references in these papers): d XO D UO dt ,

d YO D VO dt ,

d UO D au dt C bu dBu .t /,

d ZO D WO dt ,

d VO D av dt C bv dBv .t /,

d WO D aw dt C bw dBw .t /.

(10.2)

Here by UO , VO , WO we denote the components of the model Lagrangian velocity, Bu .t /, Bv .t /, Bw .t / are three standard independent Wiener processes; au , av , aw and bu , bv , O YO , Z, O UO , VO , WO /, in general. bw are functions of .t , X, Ideally, one would have an approximation such that the true and the model Lagrangian velocities coincide: O /, t /, UO .t / D u.XO .t /, YO .t /, Z.t O /, t /, VO .t / D v.XO .t /, YO .t /, Z.t

O /, t /, WO .t / D w.XO .t /, YO .t /, Z.t

(10.3)

which would assure that the true and the model trajectories are the same. However it is unrealistic to satisfy .10.1/, and therefore one uses different consistency principles. The general consistency principle says that the statistics of the model process O /, UO .t /, VO .t /, WO .t / satisfies the same relations satisfied by the true XO .t /, YO .t /, Z.t process X.t /, Y .t /, Z.t /, U.t /, V .t /, W .t /, where U.t / D u.X.t /, Y .t /, Z.t /, t /, V .t / D v.X.t /, Y .t /, Z.t /, t /, W .t / D w.X.t /, Y .t /, Z.t /, t / are the components of the true Lagrangian velocity. Two consistency principles used in the literature are: (a) consistency with the Kolmogorov similarity theory; (b) consistency with Thomson’s well-mixed condition. Here (a) reads h.d U /2 i D h.d V /2 i D h.d W /2 i D C0 "dt , and hd U d V i D hd U d W i D hd W d V i D 0, where d U , d V , d W are the components of the increments of the Lagrangian velocity, " is the mean rate of the dissipation of turbulence energy, C0 is the universal constant (e. g., [145, 215, 237]); here and in what follows, the angle brackets stand for the ensemble average. Note p that (a) implies (e. g., see [237]) that in .10.1/ all the terms bu , bv , bw are equal to C0 ": p (10.4) bu D bv D bw D C0 ".

195

Section 10.1 Introduction

The Thomson well-mixed condition can be rigorously derived from Novikov’s integral relation [153] Z pL .x, y, z, u, v, w; x0 , y0 , z0 , t /dx0 dy0 dz0 . (10.5) pE .u, v, w; x, y, z, t / D R3

Here pE is the probability density function (p. d. f.) of the Eulerian velocity u, v, w, in the fixed point x, y, z, at the time t , and pL is the joint p. d. f. of the true Lagrangian phase point X , Y , Z, U , V , W defined by the trajectory started at x0 , y0 , z0 . It is natural to require that the p. d. f. of the model phase point governed by .10.1/, say pOL , satisfies Z pE .u, v, w; x, y, z, t / D

R3

pOL .x, y, z, u, v, w; x0 , y0 , z0 , t /dx0 dy0 dz0 .

(10.6)

Note that .10.6/, the Focker–Planck–Kolmogorov equation for pOL and .10.4/ lead to the well-mixed condition due to Thomson [237]: @ @ @pE @pE @pE @pE @ Cu Cv Cw C .au pE / C .av pE / C .aw pE / @t @x @y @z @u @v @w ² ³ C 0 " @2 pE @2 pE @2 pE C C . (10.7) D 2 @u2 @v 2 @w 2 It is convenient to rewrite this equation in the form given by Thomson [237], @ @ @pE @pE @pE @ @pE Cu Cv Cw C .u / C .v / C .w / D 0, (10.8) @t @x @y @z @u @v @w where u D au pE 

C0 " @pE , 2 @u

v D av pE 

C0 " @pE , 2 @v

w D aw pE 

C0 " @pE . 2 @w

The vector function  D .u , v , w / is not uniquely defined from .10.8/. Indeed, a series of solutions can be obtained by adding to  an arbitrary vector function whose divergence in velocity space is zero. It should be noted that in the 1-dimensional case, the well-mixed condition uniquely defines the LS model even for non-Gaussian pE [237]. In the multidimensional case, the uniqueness problem can be formulated as follows: give physically plausible assumptions which define uniquely the function  in .10.8/. The first 3D LS model satisfying .10.8/ was suggested by Thomson [237]. The p. d. f. pE in his model has a Gaussian form, and the drift terms au , av , aw have a quadratic dependence on the velocity. Another example of a model (suggested by

196

Chapter 10

Dispersion in atmospheric boundary layer

Borgas; see, e. g., [255]) with a quadratic drift term and Gaussian pE was studied in [217]. They found that the Borgas model gives slightly different results compared to Thomson’s model. Reynolds [177] has constructed a 2-parameter class of wellmixed models, also quadratic and with Gaussian pE , which includes Thomson’s and Borgas’ models. He demonstrated that two different models from his class produce essentially different predictions of the turbulent dispersion. The non-Gaussian form of pE in the 2D case was treated in [56]. To extract a unique model in the class of well-mixed models, they suggested the following ad hoc assumption: the term .u =pE , w =pE / accelerates particles directly towards (or away from) the origin of .u, w/ space. A 3D generalization of this model is given by Monti and Leuzzi [147]. A further generalization of the approach of Flesch and Wilson [56] was given in [255]: the vector .u =pE , w =pE / is chosen so that there is no preferred direction of rotation of the velocity fluctuation vector (“zero-spin” models). However as shown by Reynolds [178], this approach does not solve the uniqueness problem. It should be emphasized that all the above-mentioned LS models deal with quite general inhomogeneous turbulence flows. It is therefore difficult to formulate physically motivated assumptions which, together with the well-mixed condition, uniquely define the LS model. Therefore it is reasonable to consider special classes of turbulent flows (e. g., horizontally homogeneous) whose specific features can be used to construct uniquely the LS models under assumptions with a credible physical basis. In the present chapter we treat a 3D horizontally homogeneous surface layer with a general form of pE and formulate a physically plausible assumption about the structure of the drift terms au , av , aw . This assumption uniquely defines our model in the class of well-mixed models. The model proposed is essentially different from all the models cited above, in particular, for Gaussian pE , our model, being in this case also quadratic in velocities, is generally not in the class of models given by Reynolds [177]; an exception concerns the case of ideally neutral stratification (i. e., the Obukhov– Monin length scale is infinite: L D 1): in this case our model belongs to Reynolds’ class if the Reynolds parameters C1 and C2 are chosen as C1 D C0 u4 =2w4 , and C2 D 0 (see Appendix B of this chapter). In Section 10.2 we formulate the assumption which ensures the unique definition of our model for the horizontally homogeneous neutrally-stratified surface layer. Comparison with other Lagrangian stochastic models and experimental measurements is given in Section 10.3. The convective case is treated in Section 10.4 . The behavior of trajectories of our model near the boundary is analyzed in Section 10.5. In Appendix A the drift terms are derived in the Gaussian case. In Appendix B we analyze how our model relates to Reynolds’ and “zero-spin” classes of models.

Section 10.2 Neutrally stratified boundary layer

10.2

Neutrally stratified boundary layer

10.2.1

General case of Eulerian p. d. f.

197

We consider a horizontally homogeneous incompressible boundary layer in the half3 D ¹.x, y, z/ : z  0º, where x, y are the horizontal coordinates and space RC z is the vertical coordinate. Thus it is assumed that the mean velocity has no vertical component. It is supposed in this section that the mean velocity vector does not change its direction with height; it is directed along the X -axis, and the crosswind velocity fluctuations are symmetric with respect to the plane XZ. Thus the mean velocity vector is .u.z, N t /, 0, 0/, while pE does not depend on x, y. We will write the p. d. f. pE in the form 0 .u0 , v 0 , w 0 ; z, t /, pE .u, v, w; z, t / D pE

where u0 D u  u.z, N t /, v 0 D v and w 0 D w. By .10.4/, equation .10.1/ in these variables has the form N t //dt , d Y D V 0 dt , dZ D W 0 dt , dX D .U 0 C u.Z, p d U 0 D au0 .t , Z, U 0 , V 0 , W 0 /dt C C0 " dBu .t /, p d V 0 D av0 .t , Z, U 0 , V 0 , W 0 /dt C C0 " dBv .t /, p 0 d W 0 D aw .t , Z, U 0 , V 0 , W 0 /dt C C0 " dBw .t /.

(10.9)

To simplify the notation, here and in what follows we omit the hat sign introduced in Section 10.1 to denote the model trajectory. The well-mixed condition in new variables is 0 @p 0 @pE @ @ @ 0 0 / C 0 .av0 pE /C .a0 p 0 / C w 0 E C 0 .au0 pE @t @z @u @v @w 0 w E ² 0 0 0 ³ @2 pE @2 pE C 0 " @2 pE D C C . 2 @u02 @v 02 @w 02

(10.10)

Now we give our main assumption about the structure of the Lagrangian model .10.9/. Assumption 10.1. We assume in addition to the well-mixed condition that (i)

the vertical drift term does not depend on the horizontal velocity components: 0 D a0 .t , z, w 0 /; aw w

(ii) the streamwise term au0 does not depend on the crosswind velocity v 0 : au0 D au0 .t , z, u0 , w 0 /.

198

Chapter 10

Dispersion in atmospheric boundary layer

This assumption meets the conditions of a surface layer with neutral (or close to neutral) stratification. Indeed, all the contributions to the vertical motions can be divided into two parts: the first comes from the vortices whose sizes are smaller or close to the current height z, and the second is due to the large horizontally stretched vortices. The second part of the contribution is much smaller than the first one, since the vertical velocities in such horizontal stretched vortices are much smaller than those of the small vortices whose sizes are of the order of the current height. The first part comes mainly from vortices which are isotropic or at least not very anisotropic. But in the 0 D a0 .t , z, w 0 / isotropic case, the well-mixed condition leads to the dependence aw w (e. g., see [256]) which gives us the motivation of point (i) in our assumption. As for point (ii), we note that the coordinate system is chosen so that hu0 v 0 i D 0, hv 0 w 0 i D 0, but hu0 w 0 i ¤ 0, which suggests the approximation au0 D au0 .t , z, u0 , w 0 /. The approximation formulated in point (ii) is reasonable if the mean velocity is dominating over the fluctuated part. Otherwise, for instance in the convective case, this approximation fails, and the velocity components u0 and v 0 must enter the drift terms symmetrically. In Section 10.3 we will treat this case separately. Note that the dependence au0 D au0 .t , z, u0 , w 0 / holds both for Thomson’s and Reynolds’ models; see Appendix B of this chapter. Thus the model .10.9/, in view of the assumption reads N t //dt , d Y D V 0 dt , dZ D W 0 dt , dX D .U 0 C u.Z, p d U 0 D au0 .t , Z, U 0 , W 0 /dt C C0 " dBu .t /, p d V 0 D av0 .t , Z, U 0 , V 0 , W 0 /dt C C0 " dBv .t /, p 0 d W 0 D aw .t , Z, W 0 /dt C C0 " dBw .t /.

(10.11)

Integrating .10.10/ over u0 and v 0 yields 0 0 @p 0 @p1E C0 " @2 p1E @ 0 0 0 .a .t , z, w /p / D , C w 0 1E C 1E @t @z @w 0 w 2 @.w 0 /2

(10.12)

where 0 0 p1E D p1E .w 0 ; z, t / D

Z

Z

1

1

1 1

0 pE .u0 , v 0 , w 0 ; z, t / du0 dv 0 .

Here we have assumed that 0 0 au0 pE , av0 pE ,

0 0 @pE @pE , @u0 @v 0

all tend to zero as

.u0 /2 C .v 0 /2 ! 1.

(10.13)

Section 10.2 Neutrally stratified boundary layer

199

Similarly, the integration of .10.10/ over v 0 leads to 0 @p 0 @p2E @ @ 0 0 /C .a0 .t , z, w 0 /p2E / C w 0 2E C 0 .au0 .t , z, u0 , w 0 /p2E @t @z @u @w 0 w  0 0 @2 p2E C0 " @2 p2E D C , 2 @.u0 /2 @.w 0 /2

where 0 0 D p2E .u0 , w 0 ; z, t / D p2E

Z

1

1

0 pE .u0 , v 0 , w 0 ; z, t / dv 0 .

(10.14)

(10.15)

Now, under the assumption about the behavior in infinity, it is possible to uniquely 0 . Indeed, from .10.12/ one gets a0 , then from define the coefficients au0 , av0 and aw w 0 .10.14/ one finds au , and from .10.10/ one obtains av0 . This yields ²  ³ 0 1 C0 " @p1E @f1E @F1E 0 aw .t , z, w/ D 0  C , (10.16) p1E .w; z, t / 2 @w @t @z where

Z f1E .w; z, t / D Z F1E .w; z, t / D

and au0 .t , z, u, w/

1 D 0 p2E

²

1 w

1

Z f2E .u, w; z, t / D

Finally, 1 D 0 pE

0 p1E .w 0 ; z, t / dw 0 ,

0 w 0 p1E .w 0 ; z, t / dw 0 ,

 0 @2 f2E C0 " @p2E C 2 @u @w 2  ³ @f2E @f2E @ 0  Cw  .a f2E / , @t @z @w w

where

av0 .t , z, u, w/

w

u

1

(10.17)

0 p2E .u0 , w; z, t / du0 .

 0 @pE @ 2 fE C 0 " @ 2 fE C C 2 @u2 @v @w 2  ³ @ 0 @fE @ @fE 0  Cw  .a fE /  .a fE / , @t @z @u u @w w (10.18)

²

where

Z fE .u, v, w; z, t / D

v

1

0 pE .u, v 0 , w; z, t / dv 0 .

200

Chapter 10

Dispersion in atmospheric boundary layer

Thus the coefficients .10.16/–.10.18/ define a unique stochastic model .10.11/ 0 . through the p. d. f. pE In the case when the crosswind velocity fluctuations are independent of the streamwise and vertical fluctuations, i. e., if 0 0 .u, v, w; z, t / D p2E .u, w; z, t /pvE .v; z, t /, pE

(10.19)

then the expression .10.18/ for the crosswind drift term can be simplified: av0 .t , z, u, v, w/ D

C0 " @pvE 1 @fvE w @fvE   , 2pvE @v pvE @t pvE @z

where

Z fvE .v/ D

v

(10.20)

pvE .v 0 / dv 0 .

1

10.2.2 Gaussian p. d. f. We present here expressions for the coefficients to .10.11/ for Gaussian p. d. f. pE . Recall that we deal here with horizontally homogeneous turbulence, and use a coordinate system where the direction of the mean velocity coincides with the X-axes, and the crosswind velocity fluctuations are symmetric relative to the plane XZ. Therefore, the Gaussian p. d. f. pE has the form 0 .u, v, w; z, t / D pE

1

² exp 

1

2 2u=w ² ³ v2 1 exp  2 , p 2v 2v

2u=w w

.u  w/2 

w2 2w2

³

(10.21)

where u=w D

1=2 , w

D

uw , w2

 D u2 w2  .uw/2 ,

and u2 , v2 , w2 are the variances of the x-, y- and z-velocity components, respectively. Note that by definition, u=w is the standard deviation of u conditional on the value of w. Using the result given in Section 2.1 we obtain (see Appendix A of this chapter) the following expressions: 0 .t , z, w/ D  aw



C0 " 1 @w  2w2 w @t

wC

1 @w2 2 @z



w2 C 1 , w2

(10.22)

201

Section 10.3 Comparison with other models and measurements

au0 .t , z, u, w/

 C0 ".1 C 2 /  @w2 D .u  w/ C 2 C0 " C w 2 2u=w 2w @t    @w2 w 2 @ @ C C1 Cw w 2 @z w2 @t @z  @u=w @u=w 1 Cw .uw/, (10.23)  u=w @t @z 

and av0 .t , z, u, v, w/

C0 " 1 @v D  2 2v v @t

vC

1 @v2 vw . 2 @z v2

(10.24)

Note that in the stationary case these expressions can be simplified to  C0 ".1 C 2 /  C0 "  @w2 w 2 0 au .t , z, u, w/ D  .u  w/ C wC C1 2 2u=w 2w2 2 @z w2 

@ 2 1 @u=w w  .u  w/w, @z u=w @z

C0 " 1 @v2 vw v C , 2v2 2 @z v2  C0 " 1 @w2 w 2 0 C1 . aw .t , z, w/ D  2 w C 2w 2 @z w2

av0 .t , z, u, v, w/ D 

(10.25)

10.3

Comparison with other models and measurements

10.3.1

Comparison with measurements in an ideally-neutral surface layer (INSL)

In this section we analyse some quantities in the case of turbulent dispersion in a stationary, horizontally homogeneous, ideally-neutral surface layer (i. e., the Obukhov– Monin length L equals infinity). We have calculated the following dimensionless Lagrangian characteristics: ´ μ hZ.t /i z0 hX.t /i hZ 2 .t /i1=2 , B.t / D , C.t / D exp C 1 , (10.26) A.t / D u t u t u t u t and the ratio pr.z/ D k .z/=kz .z/, where is the von Karman constant, k D u z is the vertical eddy viscosity, and kz is the vertical eddy diffusivity coefficient defined through the Boussinesque hypothesis c 0 w 0 .z/ D kz .z/

@c.z/ N . @z

(10.27)

202

Chapter 10

Dispersion in atmospheric boundary layer

The importance of the characteristics A.t /, B.t /, and C.t / is that these functions tend, as t ! 1, to some universal constant values a, b, and c, respectively, provided that hs and z0 are much less than u t (e. g., see [19, p. 77]). Here hs is the height at which the Lagrangian trajectory starts. As to the ratio pr.z/, for values z much larger than the source height, it tends to the Prandtl constant P r; its universal character is well known and is in the literature often approximately taken equal to unity (e. g., see [145, Sect. 8.2]). Since all the four quantities A.t /, B.t /, C.t /, and pr.z/ do not depend on the crosswind dispersion, we use the 2D stochastic models to simulate the dispersion: N t //dt , dZ D W 0 dt , dX D .U 0 C u.Z, p d U 0 D au0 .Z, U 0 , W 0 /dt C C0 " dBu .t /, p 0 d W 0 D aw .Z, U 0 , W 0 /dt C C0 " dBw .t /,

(10.28)

where for the ISNL, the vertical profiles of " and uN can be taken as (e. g., see [145]) ".z/ D

u3 , z

u.z/ N D

 u  ln z=z0 / ,

and z0 is the roughness height. The calculations were carried out by Thomson’s, Reynolds’, Flesch and Wilson’s, and our models. Thomson’s 2D model in this case is specified by  C0 ".z/  2 w u C u2 w , au0 .z, u, w/ D  2 0 .z, u, w/ D  aw

 C0 ".z/  2 u w C u2 u , 2

where u and w are given by u D bu u , w D bw u with u2 D uw D const ;  D u2 w2 uw 2 , bu and bw are universal constants. Following Panofsky and Dutton [161], and Stull [232] we have taken bu D 2.5, bw D 1.25. These parameters enter all the models specified below. The drift terms of the model due to Flesch and Wilson [56] can be written in the case of INSL as au0 .z, u, w/ D 

0 C0 ".z/ @ ln pE C0 ".z/ D  2 .u  w/, 2 @u 2u=w

0 .z, u, w/ D  aw

0 C0 ".z/ @ ln pE C0 ".z/ C0 ".z/ .u  w/  w, D 2 2 @w 2u=w 2w2

p where u=w D =w ,  D uw=w2 . It turns out that the Thomson and the Flesch and Wilson models are identical for the case of INSL. This can be seen directly by comparing of the coefficients.

Section 10.3 Comparison with other models and measurements

203

The model of Reynolds [178] in our case of ideally-neutral surface layer is specified by au0 .z, u, w/



C0 " d uN D C C1 u2 2 dz

0 aw .z, u, w/ D 



w2 u C u2 w , 

 C0 ".z/  2 d uN w2 u C u2 w u w C u2 u C C1 w2 , (10.29) 2 dz 

with C1 D 3 chosen by Reynolds [178] to fit the experimental results of Legg [127]. Our model .10.22/–.10.23/ (in the following we call it a KS model) in the case of INSL is specified by au0 .z, u, w/ D  0 .z, w/ D  aw

C0 ".z/.1 C 2 /  C0 ".z/ .u  w/ C w, 2 2u=w 2w2 C0 ".z/ w. 2w2

(10.30)

4 , then it reduces Note that if we choose the parameter C1 in .10.29/ as C1 D C0 =2bw to our model .10.30/ (see Appendix B of this chapter). Reynolds however suggests in his model C1 D 3, and in all comparisons below, when referring to Reynolds’ model, we take C1 D 3. In all models, the calculations were carried out for z0 D 0.01 m, the trajectories started at hs D 0.02 m, u D 0.4 m s1 , the number of trajectories was N D 105 in the case of a, b, and c calculations, and N D 106 for the constant P r. The vertical eddy diffusivity kz was calculated from the relation .10.27/ where a finite-difference approximation of the calculated mean concentration was used to find the mean concentration derivative. A stationary source was uniformly distributed on the plane z D zs D 0.02 m. The stochastic differential equations were solved by the explicit Euler scheme, with the varying time step t D ˛L .z/, where L .z/ D 2w =C0 ".z/ is the Lagrangian time scale; to reach stable numerical results, we found that ˛ D 0.02 was sufficient. At the boundary, a perfect reflection is made after the trajectory hits the layer ¹z < z0 º, z0 being the roughness height. The results of calculations and experimental data are presented in Table 10.1. The calculations were carried out for different values of C0 , since the constant C0 is known to be scattered in the interval .2, 8/ (e. g., see [170]). The results for all four constants a, b, c, and P r show that our model is in a good agreement with Thomson’s model and experimental results, but in poor agreement with the results obtained by Reynolds’ model. We recall that a, b, and c, as defined above, are universal constants determined as limits, as t ! 1, of the functions A.t /, B.t /, and C.t /. In calculations, t was taken sufficiently large. Note that the Thomson and the Flesch and Wilson models (which are identical in this case) give slightly different values of pr, while the constants a, b, and c practically coincide. This can be explained by the fact that in the case of pr,

204

Chapter 10

Dispersion in atmospheric boundary layer

Table 10.1. Universal constants a, b, c and P r calculated by different Lagrangian models, compared against experimental results. Model

C0

a

b

c

Pr

Thomson 1987 [237]

3. 4. 5. 7.

0.85 0.71 0.61 0.48

0.65 0.54 0.46 0.35

0.25 0.22 0.2 0.16

0.47 0.6 0.74 1.

Flesch and Wilson 1992 [56]

3. 5. 7.

0.85 0.61 0.48

0.65 0.46 0.35

0.26 0.2 0.16

0.45 0.8 0.9

KS (see .6.14/, Section 3.1)

3. 4. 5. 7.

0.73 0.59 0.5 0.37

0.55 0.44 0.36 0.27

0.17 0.15 0.14 0.11

0.64 0.82 1. 1.43

Reynolds 1997 [177]

3. 5. 7.

0.13 0.18 0.21

0.09 0.13 0.15

0.04 0.06 0.07

5.26 3.3 2.86

0.58

0.44 0.4

0.19

Measurements Garger and Zhukov 1986 [64] Chaudhry and Meroney 1973 [21] Rider 1954 [179] Gurvich 1965 [73]

0.83 1.25

additional error of the finite-difference approximation to the concentration derivative in .10.27/ is involved. As to the best fit to the experimental data, our model reaches it at C0 D 4, while Thomson’s and Flesch and Wilson’s models fit best at C0 D 5. Regarding the Reynolds model, it should be mentioned that as C0 becomes larger, the discrepancy between the results obtained by his model and measurements slightly decreases, but even for C0 D 7 it remains too large. Calculations for C0 D 10 (not shown in the table) gave almost the same results as for C0 D 7.

10.3.2 Comparison with the wind tunnel experiment by Raupach and Legg (1983) In this section we present a comparison of the same models analysed in the previous section against the data of the wind tunnel experiment by Raupach and Legg [176]. The vertical profiles of the mean concentration c, N the streamwise and vertical fluxes of concentration c 0 u0 , c 0 w 0 , were analysed. A stationary line source at the height hs D 0.06 m directed along the y-axis was considered, and all the profiles were calculated

Section 10.3 Comparison with other models and measurements

z= hs

205

?sbb 33 ?b 3 ?ssb 3 Raupach & Legg experiment   ?sb 2.5 3 Thomson’s model ı ? bss 3 Reynolds’ model ˘ 3? b s KS model ? s 3? s b 2 ? sb 3 3? s sb 3? s s3 ?b s 1.5 b s?3 s ? b s s3? b 3 1 ss b ?b 3 3 s b ?3 b s? s ? 3 0.5 3 s ?b s ?b 3 s sb ? 03 0 0.2 0.4 0.6 0.8 1 dimensionless concentration c=c N 

z= hs

?sbb 33 ?b 3 ?ssb 3 Raupach & Legg experiment   ?bs 3 2.5 3 Thomson’s model ı ? bss Reynolds’ model ˘ ? 3b s KS model ? ? bs s 3 2 3 ? bs s ? 3 ?bs s 3 s? s b 3 1.5 b s ? 3s ? b s s3 ? bb 3 1 ? 3 ss ?b 3 s b ?3 s b 0.5 s ?s b s 3 ? 3 3b ?sb s ? 3 0 0 0.2 0.4 0.6 0.8 1 1.2 dimensionless concentration c=c N 

Figure 10.1. A comparison of three model predictions of vertical profile of mean concentrtaion with Raupach and Legg’s measurement, for C0 D 3 (upper panel) and C0 D 7 (lower panel).

at the downwind distance x D 7.5 hs . The problem is governed by the 2D equations used in the previous subsection. In Figure 10.1, upper panel, the scaled mean concentration c.x, N z/=c and temperN z/= profiles are plotted for C0 D 3, where ature .x, N s //, c D Q=.hs u.h

 D Q=.cp hs u.h N s //.

Here Q is the line source strength per unit length,  the air density and cp the specific heat of air at constant pressure. The temperature profiles were taken from the mea-

206

Chapter 10 3 2.5 2 z= hs 1.5 1 0.5

?b ?b ?b

0 0.4

Dispersion in atmospheric boundary layer

?sb 3 ?b 3 3?b bss 3? b s 3? bs s 3 ? bs 3 ? s sb 3 ? s b 3 ?s s b 3 ?s b 3ss ?b 3? s ?b b3s s 3 ? s ?b 3 s 3s ? sb Raupach & Legg experiment   3 sss Thomson’s model ı 3 Reynolds’ model ˘ 3 s b ? s 3 KS model ? ?bs 3 0.2 0 0.2 0.4 0.6 dimensionless vertical flux w 0 c 0 =u c

3 2.5 2 z= hs 1.5 1

?sb 3 ?b 3 ?b ss 3 3?bb s 3? b s s 3? b ss 3? 3 ? b sb s s 3 ? 3 ? b s sb 3 ? ss b s?b3 b3s s 3? ? s s ?b3

b? 3 s s ?b 3 s b ? ss 3 0.5 ?bs 3 sb ?s3b ? 3 0 0.4 0.3 0.2 0.1 0

Raupach & Legg experiment Thomson’s model Reynolds’ model KS model

0.1

0.2

dimensionless vertical flux

0.3 w 0 c 0 =u

0.4

  ı ˘ ?

0.5

 c

Figure 10.2. A comparison of three model predictions of the vertical profile of mean vertical flux with Raupach and Legg’s measurement, for C0 D 3 (upper panel) and C0 D 7 (lower panel).

surements by Raupach and Legg [176] where the temperature was the tracer in the experiments. All the models predict the experimental results qualitatively well. We mention that the results obtained by Flesch and Wilson’s model are very close to the results obtained by Thomson’s model, and therefore we do not plot them in our figures. Above the height z D 1.75hs all three models agree well with the experimental results. Below the height 1.5hs the Thomson and the KS models give results close to the measurements, while Reynolds’s model overestimates the maximum concentration and under-

207

Section 10.4 Convective case

estimates the concentration near the ground. As to the sensitivity to the constant C0 , we have also made calculations for C0 D 2, 4, 5, and 7. The best fit of Thomson’s and KS models was found at C0 D 3. For larger values of C0 (see Figure 10.1, lower panel for C0 D 7) all the models overestimate the values at the maximum, and underestimate them at small and large heights. In Figure 10.2 the vertical profile of the vertical flux of concentration is shown for C0 D 3. From these curves, it is clearly seen that at the height z < zs the Thomson and the KS models underestimates, and Reynolds’ model overestimates the experimental results. Above the height z D 1.5zs all three models are in a good agreement with the measurements. Note that for C0 D 7 the picture is different (see the lower panel of Figure 10.2): the models give slightly better predictions for heights z < 1.5zs In Figure 10.3 the vertical profiles of the streamwise flux of concentration are presented for C0 D 3 (upper panel) and C0 D 7 (lower panel). Here the Reynolds model significantly overpredicts the maximum and underpredicts the minimum values. the Thomson and the KS models show better agreement with the measurements. Note, however, that the agreement between the Thomson and KS models in this case is not so good as in Figures 10.1 and 10.2. Calculations of the vertical and horizontal fluxes by our model with different values of C0 have shown that the best fit with the Raupach and Legg experimental results was around C0 D 3.5 ˙ 0.5. This value is in good agreement with the value C0 D 3 suggested in [43].

10.4

Convective case

In this section we consider a horizontally homogeneous boundary layer under strong convective conditions at sufficiently large heights compared to jLj. In this case, the velocity fluctuations can be considered as horizontally isotropic (e. g., see [145]). Therefore, the mean velocity is zero, and the correlation between the vertical and horizontal velocities is zero. In this section we show that the horizontal isotropy and the dependence supposed in (i) of the assumption ensure the unique choice of the Lagrangian stochastic model for the convective layer. To construct the Lagrangian 1-particle model in the convective case, we have to specify the Eulerian velocity p. d. f.. For simplicity, we will treat the case when the Eulerian p. d. f. has the form k

? .u? ; z, t /, pE .u, v, w; z, t / D pE .w; z, t /pE

(10.31)

p k ? where u? D u2 C v 2 , pE is the p. d. f. of the vertical velocity component, and pE is the p. d. f. of the horizontal velocity components satisfying the relation Z 1 Z 1 Z 1 p ? ? 2 2 du dvpE . u C v ; z, t / D 2 du? u? pE .u? ; z, t / D 1. 1

1

0

208

Chapter 10

Dispersion in atmospheric boundary layer

?bs 3 ?bb 3 ?s 3 Raupach & Legg experiment ?ssb 3 2.5 b Thomson’s model s ? 3 s Reynolds’ model ?b 3 s b s KS model 3? s b 2 3 b ?s ?b ss 3 z= hs 3 ?b s ? b ss 1.5 3 ? 3 ? b s sb 3 ? bs s 3 1 ?3 b s s b? s?s 3 3 b s 3 0.5 b s s ?? 3 3sb ? 3 sb ? 0 1 0.5 0 0.5 1 3

  ı ˘ ?

dimensionless horizontal flux u0 c 0 =u c ?bs 3 ?bb 3 s ? Raupach & Legg experiment   s3 b s? 3 Thomson’s model ı 2.5 b s ? 3 s?b Reynolds’ model ˘ 3 s KS model ? s? 3b 2 ?ssb 3 b ?s 3 z= hs 3 b?b ss ? 3 1.5 b ss ? 3 ss b ? 3 ? b bs s 3 1 3? s s b s 3 b? b ss ?? 3 3 0.5 b ?s b ?s s 3 3 b s 3? 0 1 0.5 0 0.5 1 1.5 3

dimensionless horizontal flux u0 c 0 =u c Figure 10.3. A comparison of three model predictions of the vertical profile of mean streamwise flux with Raupach and Legg’s measurement, for C0 D 3 (upper panel) and C0 D 7 (lower panel).

Note that in the convective case, the assumption .10.31/ is quite reasonable, because the vertical and horizontal velocity fluctuations can be considered as approximately independent.

209

Section 10.4 Convective case

Under the horizontally isotropy and assuming that the vertical velocity component is governed as assumed in the assumption, point (i), the model .10.9/ takes the form dX D Udt ,

d Y D Vdt ,

dZ D W dt , p d U D Ug.t , Z, U? , W /dt C C0 "dBu .t /, p d V D Vg.t , Z, U? , W /dt C C0 "dBv .t /, p d W D aw .t , Z, W /dt C C0 "dBw .t /.

(10.32)

Thomson’s well-mixed condition implies in our case that u?

@ @ @ @pE .u? , w; z, t / .u2? g pE / C C .wu? pE / C .u? aw pE / @t @z @u? @w ³ ²  C0 " @ @pE @2 .u p / D u? C ? E . (10.33) 2 @u? @u? @w 2

This relation follows from .10.7/ and from the following simple equalities: @ @ 1 @ .u2 gpE /, .ugpE / C .vgpE / D @u @v u? @u? ? @2 pE @2 pE 1 @  @pE  @2 pE @2 pE u? C C C D . @u2 @v 2 @w 2 u? @u? @u? @w 2 The well-mixed condition .10.33/ can be simplified as follows. Integrate .10.33/ R1 k over u? and use the relation 2 0 pE .u? , w; z, t /u? du? D pE .w; z, t /. Assum@pE tend to zero as u? ! 1, this yields ing u2? pE and u? @u ? k

k

@pE @ C 0 " @2 pE @ k k . C .wpE / C .aw pE / D @t @z @w 2 @w 2

(10.34)

This is the 1-dimensional well-mixed condition .10.12/. As in the case .10.12/, we can find from .10.34/ the coefficient aw .t , z, w/ aw .t , z, w/ D

1 k

pE .w; z, t /

´

k

C0 " @pE  2 @w

where

Z f1E .w; z, t / D

1

Z F1E .w; z, t / D

w

w

1



@f1E @F1E C @t @z

k

pE .w 0 ; z, t / dw 0 , k

w 0 pE .w 0 ; z, t / dw 0 .

μ ,

(10.35)

210

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Dispersion in atmospheric boundary layer

This form agrees with that used by Luhar and Britter [133], who explored the value of aw in detail for a particular choice of pE . To find the function g, we substitute .10.31/ in .10.33/, which in view of .10.34/ yields  ? ? @pE @pE @ C0 " @ @ ? 2 ? .u g p / D C .wu? pE / C u? . (10.36) u? @t @z @u? ? E 2 @u? @u? ? ! 0 as Integrating .10.36/ over u? from 0 to 1 we get, under the condition u2? gpE u? ! 0, that ? C0 " @pE @PE @PE ? D , Cw C u2? g.t , z, u? , w/pE u? @t @z 2 @u?

where

Z PE .u? ; z, t / D

0

u?

(10.37)

? upE .u; z, t /du.

? is given. This defines the function g if pE For example, if ? .u? / pE

² ³ u2? 1 D exp  2 , 2 2 .t , z/ 2 .t , z/

 ² ³ u2? 1 1  exp  2 , PE .u? / D 2 2 .t , z/ then @PE @ ln ? 2 ? D .u? pE /, @t @t

@ ln ? 2 ? @PE D .u? pE /, @z @z

? @pE u? ? D  2 pE , @u? ?

and we get gD

1 ? u2? pE

D

²

? @PE @PE C0 " @pE  u? w 2 @u? @t @z

³

C0 " @ ln ? @ ln ? w . 2  2? @t @z

Remark 10.1. We have assumed here the factorization .10.31/, which simplifies the form of g. Generally, when .10.31/ is not true, the function g can be found analogously, but its structure is more complicated.

211

Section 10.5 Boundary conditions

10.5

Boundary conditions

To complete the description of the Lagrangian stochastic model, we need to define the behavior of .X.t /, Y .t /, Z.t /, U.t /, V .t /, W .t //, the solution to .10.1/ in the neighborhood of the boundary D ¹.x, y, z/ : z D 0º. We assume that the boundary is impenetrable, i. e., that w D 0 at the boundary of . This implies that the true Lagrangian trajectories never reach . Therefore it is reasonable to require that the same property holds for X.t /, Y .t /, Z.t /, the solutions to .10.1/. This can be done by special choice of the function ".z, t /. Indeed, in the neighborhood of , it is reasonable to consider the flow as ideally neutral stratified. Therefore, pE .w/ is Gaussian, with constant w , and the vertical profile of ".z/ is given by (e. g., see [145]) ".z/ D

u3 , z

' 0.4,

z > z0 .

(10.38)

Here is the Karman constant, and z0 is the roughness height. The equation of vertical motion then is dZ D W dt ,

d W .t / D 

b a W .t /dt C p dB.t /, Z Z

where aD

u3 , 2 w2

 bD

C0 u3

(10.39)

12 .

If we assume that the formula .10.38/ is true for all z > 0, then all the solutions to .10.39/ do not reach the boundary . Indeed, let  be a random variable (which depends on the trajectory Z.t /) defined by Z t ds .  .t / D Z.s/ 0 Then, the vertical velocity in new variable W . / satisfies the equation d W . / D aW . /d  C b dB. /. Therefore, from

we have

dZ dt dZ D D W . /Z. / d dt d  Z W . 0 / d  0 . Z. / D Z.0/ exp¹S. /º, S. / D 0

The function W . / is an Uhlenbeck–Ornstein process with continuous samples. Therefore, jS. /j < 1 for all  > 0 with probability 1. This implies that Z. / > 0, provided that Z.0/ > 0. Thus the function Z. / never reaches the boundary . The same is true

212

Chapter 10

Dispersion in atmospheric boundary layer

for Z.t /. To show this, it is sufficient to note that t . / ! 1 as  ! 1. Let us show this property. We have Z Z Z dt 0 0 0 d D Z. / d  D Z.0/ exp¹S. 0 /º d  0 . t . / D 0 0 d 0 0 In [108] it is shown that with probability 1 Z 1 exp¹S. /º d  D 1. 0

This implies that with probability 1 t . / ! 1 as  ! 1.

10.6 Conclusion A uniquely-defined Lagrangian stochastic model in the class of well-mixed models was constructed from physically plausible assumptions: (i) in the neutrally-stratified horizontally homogeneous surface layer, the vertical motion is mainly controlled by eddies whose size is of the order of the current height, and (ii), the streamwise drift term is independent of the crosswind velocity fluctuations. Supposition (i) is motivated by the well-known property that the vertical motion of vortices whose size is much larger than the current height is damped by the ground surface. Therefore, it is reasonable to assume that the vertical drift term is the same as in the isotropic case: 0 D a0 .t , z, w/. With regard to point (ii), it comes from the assumption that in the aw w special coordinate system where the X -axis is oriented along the mean velocity vector, the crosswind velocity fluctuations are symmetrically distributed with respect to the plane XZ. In the free convective layer the mean velocity vector vanishes, and the horizontal motion is isotropic. This property is used to define uniquely the model using only e point (i) of the assumption. In the model presented the Eulerian p. d. f., pE may be not Gaussian, as, for instance, in the forest canopy [255]. The Gaussian case was analyzed in detail. The model was compared against the wind tunnel experiment of Raupach and Legg [176] and models due to Thomson [237], Flesch and Wislon [56], and Reynolds [178]. Numerical experiments have shown good agreement of our model with the models of Thomson [237], Flesch and Wilson [56] , and with experimental measurements as well. However there is a large discrepancy of these results with the results obtained by Reynolds’ model. Our model shows the best fit to the measurements for C0 D 3.5 ˙ 0.5; namely, at C D 4, we found the best agreement between the calculated and measured values of the universal constants a, b, c, and P r; at C0 D 3, the best agreement with the wind-tunnel experiments by Raupach and Legg [176] was achieved. It is interesting to note that our model, also being quadratic in velocity (in the Gaussian case), does not belong to the general 2-parameter class of models suggested by Reynolds (1997); it is also not in the family of “zero-spin” models introduced by Wilson and Flesch [255].

213

Section 10.7 Appendices

It is believed that the model proposed is well suited for the case of a neutrally (or close to neutrally) stratified surface layer. This generalization might be possible for the whole boundary layer with the mean velocity vector varying with the height, but this needs special study. The same is true for the generalization to the stably stratified surface layer where buoyancy is important.

10.7

Appendices

10.7.1

Appendix A. Derivation of the coefficients in the Gaussian case

Here we derive the coefficients .10.22/–.10.24/ from .10.16/–.10.18/ in the case of Gaussian p. d. f. .10.21/. First we remark that from .10.13/ and .10.15/ it follows ² ³ 1 w2 0 .w; z, t / D p exp  2 , p1E 2w 2w ² ³ 1 1 w2 0 2 exp  2 .u  w/  2 . p2E .u, w; z, t / D 2u=w w 2u=w 2w Consequently, Z f1E .w; z, t / D

w w

1

 w 1 2 p exp .t =2/ dt D ˆ , w 2

0 .w; z, t /. F1E .w; z, t / D w2 p1E

Note that 0 w 1 @p1E D 2, 0 p1E @w w



1 @F1E @ 2 1 D .w 2 C 1/ w , 0 p1E @z 2 @z

and w @w P @f1E D 2 ˆ.w=w /, @t w @t where

Z ˆ. / D

1

1 p exp .t 2 =2/ dt , 2

P / D dˆ . ˆ. d

From .10.16/ we find 0 .t , z, w/ D  aw



C0 " 1 @w  2w2 w @t

wC

1 @w2 2 @z



w2 C 1 . w2

Note that this coincides with Thomson’s relevant expression in his 1D model.

214

Chapter 10

Dispersion in atmospheric boundary layer

By the definition, f2E .u, w, z, t / D

0 p1E .w; z, t /



u  w ˆ . u=w

To find au0 from .10.17/ we need the expressions for @f2E , @t

@f2E , @z

@f2E , @w

0 @p2E , @u

@2 f2E . @w 2

By definition we get 0 @p2E .u  w/ 0 p2E , D 2 @u u=w  ³ ² @ 1 @w2 w 2 @f2E  1 C ‰./ D f2E , @t 2w2 @t w2 @t  ³ ² @ @f2E 1 @w2 w 2  1 C ‰./ D f2E , @z 2w2 @z w2 @z ³ ² w @f2E D f2E  2  ‰./ , @w w ³ ² h i2 w 1 @2 f2E 2P  D f  ‰./  C  ‰./ , 2E @w 2 w2 w2

where ‰. / D

d P / D d ‰. / ,  D u  w ,  D uw . ln ˆ. /, ‰. d d u=w u=w w2

Substituting these expressions in .10.17/ yields C0 " .u  w/ 2 2u=w ² ³ 0 @f2E C0 " @2 f2E 1 @aw @f2E 0 @f2E w  f2E  aw C C 0  p2E @t @z @w @w 2 @w 2  ² C0 " f2E @ 1 @w2 w 2 D  2 .u  w/ C 0  1  ‰./  2 2u=w p2E 2w @t w2 @t   



0 @ @aw 1 @w2 w 2 w 0  1 C ‰./  ‰./   a w  w 2w2 @z w2 @z @w w2 2

 ³ C0 " 1 w P C  2  ‰./  2 C 2 ‰./ . (A1) 2 w w

au0 D 

215

Section 10.7 Appendices

Since

f2E ‰ D u=w , 0 p2E

P ‰./ D ‰./. C ‰.// ,

we find from (A1) au0 .t , z, u, w/

 C0 ".1 C 2 /  @w2 D .u  w/ C 2 C0 " C w 2 2u=w 2w @t    @w2 w 2 @ @ C C 1  u=w Cw 2 @z w2 @t @z    @w2  @w2 w 2 C0 ".1 C 2 / .u  w/C " C C1 wC D C 0 2 2u=w 2w2 @t 2 @z w2   @u=w @u=w @ @ 1  Cw w Cw .uw/, @t @z u=w @t @z

Since for the case considered the condition .10.19/ is satisfied, we use here the expression .10.20/. Substituting 1 @fvE v v @v 1 @fvE v @v 1 @pvE D  2, D , D pvE @v v pvE @t v @t pvE @z v @z into .10.20/, we get av0 .t , z, u, v, w/

10.7.2

 C0 " 1 @v 1 @v2 vw D  . v C 2v2 v @t 2 @z v2

Appendix B. Relation to other models

2-parameter class of models due to Reynolds Here we analyse Reynolds’ 2-parameter class of models in the case of horizontally homogeneous turbulence with the mean velocity direction not varying with height. It is also assumed that the X -axis is oriented along the mean velocity vector, and that the crosswind velocity fluctuations are symmetric with respect to the plane XZ. Then the 2-parameter class of models quadratic in velocity, which satisfies the well-mixed condition for Gaussian pE , considered by Reynolds (1997), reads N 3 , t //dt , dX2 D U20 dt , dX3 D U30 dt , dX1 D .U10 C u.X p d U10 D a10 .t , X3 , U10 , U20 , U30 /dt C C0 " dB1 .t /, p d U20 D a20 .t , X3 , U10 , U20 , U30 /dt C C0 " dB2 .t /, p d U30 D a30 .t , X3 , U10 , U20 , U30 /dt C C0 " dB3 .t /,

.B1/

216

Chapter 10

Dispersion in atmospheric boundary layer

where C0 " 1 @i3 1  @i3 @km  ij uj C C C2 C i3 km 2 2 @z 2 @z @z  @uN 1 @j m @uN  uj C C1 i3 j 1  ı1i ı3j uj  im 2 @t @z @z @j k 1 1 @km  C2 i3 uj uk  .1  C2 /im u3 uk , 2 @z 2 @z i D 1, 2, 3. (B2)

ai0 .t , z, u1 , u2 , u3 / D 

Here we adopt the summation convention, therefore the notation .X , Y , Z/ D .X1 , X2 , X3 / and .U , V , W / D .U1 , U2 , U3 / is used; ıij is the Kronecker symbol, ij D .1 /ij are the velocity covariances which in the case considered have the form 11 D u2 ,

22 D v2 ,

13 D 31 D uw;

3 D w2 , 12 D 21 D 23 D 32 D 0, 11 D

w2 , 

22 D

1 , v2

33 D

u2 , 

uw ,  where u2 , v2 , and w2 are the variances of velocity components,  D u2 w2  .uw/2 . Thus the model includes two free parameters C1 and C2 . Reynolds (1998) has suggested C1 D ˙3, and C2 D 0 to fit the experimental results for the wind tunnel boundary layer by Legg (1983). It is interesting to see whether or not there are some values of C1 , C2 such that the model (B2) reduces to our model .10.22/–.10.24/. To this end, it is sufficient to check if the model (B2) satisfies the assumption of our model (see Section 2.1). It is clear that point (ii) of the assumption is satisfied iff C2 D 0, since in the expression for a10 , the dependence on u2 can be eliminated only if C2 D 0. Thus, taking C2 D 0, we analyse point (i) of the assumption. In the expression for a30 .t , z, u1 , u2 , u3 / we are interested in the terms depending on u1 and u2 , which ee therefore we write as 12 D 21 D 23 D 32 D 0,

a30 D ¹   º C C1 33

13 D 31 D 

1 1 @uN @1m 11 u1  C0 "31 u1  3m u3 u1 , @z 2 2 @z

where ¹   º stands for the terms not depending on u1 and u2 . From this we see that if the term @@z1m is not equal to zero, then point (i) cannot be satisfied. Note that this term is zero in the ideally-neutral stratification (L D 1). In this case (i) is satisfied, iff C1 D C0 u4 =.2w4 /. From this we conclude that only in the case of ideally-neutral stratification does our model belong to the class of models (B1)-(B2), and, for this case, the values of C1 and C2 corresponding to our model are C2 D 0, and C1 D C0 u4 =.2w4 /.

217

Section 10.7 Appendices

The “zero-spin” property Here we show that in our model (for simplicity we consider the stationary turbulence) the average increment hd ; zi to the orientation  D arctan .w 0 =u0 / of the Lagrangian velocity fluctuation vector in 2D case is negative, and hence it does not belong to the “zero-spin” class of models. Flesch and Wilson [56] show that Z 1Z 1 0  w 0 uw u du dw, hd ; zi D dt 2 C w2 u 1 1 where

0 C0 " @ ln pE u0 0 , 0 D au  pE 2 @u

0 0 C0 " @ ln pE w 0 . 0 D aw  pE 2 @w

For our model (see .10.25/) we find 1 d C0 "2 u0 2 d .u  w/ C w.u  w/  0 Dw pE dz  dz 2 2  C0 "  dw2 w2 C w C 1 C , 2w2 2 dz w2  0 1 dw2 w2 C0 " w D .u  w/, 1 C  0 2 pE 2 dz w 2 2 where we use the notation  D u=w . After some algebra we can find that hd ; zi D  dt

C0 " 2

Z

1

1

Z

1

1

 2 0 pE 2 w 2 w2 u C C du dw > 0. u2 C w 2  2 2 w2

Chapter 11

Analysis of the relative dispersion of two particles by Lagrangian stochastic models and DNS methods Stochastic Lagrangian models for the relative dispersion of two particles in a stationary, spatially isotropic incompressible, fully developed turbulent flow are studied. Along with a review of existing models, we suggest new models and compare the results with the DNS data.

11.1 Introduction Turbulent dispersion of a contaminant, for example pollutant dispersion in the atmosphere, is conveniently described in terms of Lagrangian statistics sampled along the paths of fluid particles. In practice, however, the Eulerian statistics sampled at fixed points in space are better known from experiments. Hence the basic problem of turbulent dispersion is to calculate Lagrangian statistics from given Eulerian statistics. Lagrangian stochastic (LS) models of turbulent dispersion address the problem by statistically characterizing particle paths from an Eulerian input. LS models formulate the time evolution of the particle coordinate and velocity in terms of stochastic differential equations. The LS models are best understood for the description of 1-particle statistics, which contain only 1-point statistical information and hence lack any information about the different scales of turbulent eddies. A modeled ensemble of single particles allows the calculation of mean concentrations, whereas an ensemble of particle pairs allows the calculation of concentration fluctuations. When we consider the motion of a pair particles, the modeling can be seen as the superposition of a relative motion and the motion of single particle, or particle centroid (e. g., see [44, 199, 238]). The relative motion reflects more directly the internal turbulent structure because of the appearance of an internal lengths (particle distance), and its description permits the introduction of concepts developed within the theory of turbulence [146]. To be more specific, we introduce the following notations. Throughout the chapter, we use the summation convention. The Eulerian velocity field is considered as a 3D random field denoted by UE .x, t / D .UE 1 .x, t /, UE 2 .x, t /, UE 3 .x, t //, whose samples are incompressible : @x@ UE i .x, t / D 0. The concentration of a conservative i passive scalar scattered by this field is governed by the transport equation @c @c.x, t / D 0, C UE i .x, t / @t @xi

t  0,

219

Section 11.1 Introduction

c.x, 0/ D S.x/ where S.x/ is the initial distribution of the concentration. In practice, the following quantities are of special interest: hc.x, t /i, the mean concentration, hUE i .x, t /c.x, t /i, the mean fluxes of concentration, and hc.x, t /c.x0 , t /i, the concentration covariance. There are two main approaches to evaluating these quantities. The first one is based on the averaging of the transport equation to extract closed equations for the quantities in question. This is the so-called closure problem, which faces well-known difficulties [145]. The second approach is based on the Lagrangian description, where the following representations are used: Z hc.x, t /i D d x0 S.x0 / p1L .x, t ; x0 /, Z Z hUE i .x, t /c.x, t /i D d vE d x0 vi S.x0 / p1L .E v , x, t ; x0 /, Z Z hc.x, t /c.x0 , t /i D d x0 d x00 S.x0 / S.x00 / p2L .x, x0 , t ; x0 , x00 /. Here p1L and p2L are the Lagrangian transition densities: p1L .x, t ; x0 / D hı.x  X.t ; x0 //i, p1L .E v , x, t ; x0 / D hı.x  X.t ; x0 //ı.E v  V.t ; x0 //i, p2L .x, x0 , t ; x0 , x00 / D hı.x  X.t ; x0 //ı.x0  X.t ; x00 //i. In these formula, the Lagrangian variables X, V are defined through @X D V.t ; x0 / D UE .t , X.t ; x0 //, @t

X.0; x0 / D x0 .

In this chapter we focus on the 2-particle models which describe the motion of two fluid particles. It is governed by d X1 .t / D UE .t , X1 /, dt

d X2 .t / D UE .t , X2 /, dt

where X1 .t / D X.t ; x0 / and X2 .t / D X.t ; x00 /. It is convenient to rewrite this system as ² ³   r  r  1 dR D UE R C , t C UE R  , t , dt 2 2 2   r  r  dr D UE R C , t  UE R  , t , dt 2 2

(11.1)

220

Chapter 11 Analysis of the relative dispersion of two particles

where

 1 X1 .t / C X2 .t / , r.t / D X1 .t /  X2 .t /. 2 This form clearly illustrates the division of large and small scales of turbulence. Indeed if the distance between the two particles is much less than the external scale (r  L), then the terms   r r UE R C , t C UE R  , t 2 2 R.t / D

  r r UE R C , t  UE R  , t 2 2 are approximately statistically independent. This property can be used to simulate the motion of two particles in small scales according to ²   ³ 1 Q r dR Q R  r,t , D U R C ,t C U dt 2 2 2 and

dri .t / D vi .t / dt ,

dvi .t / D ai .r, vE, t / dt C bij .r, vE, t / d Wj .t /,

i D 1, 2, 3.

Here the large scale velocity field is approximated by a field marked by the tilde (e. g., extracted from DNS or LES methods), while the small scale motion is described as a diffusion process governed by a Langevin type equation. Two alternative modelling approaches include Eulerian statistics in LS models: the well-mixed approach of Thomson [237] and the moments approximation method of Novikov [155]. The importance of Thomson’s approach is that when the material distribution is uniform, the model does not artificially unmix the material. A one dimensional well-mixed LS model of relative dispersion of two particles has been proposed by Thomson [236]. 3-dimensional models based on well-mixed criteria have been considered in Thomson [15, 238]. Gaussian Eulerian statistics are used in these articles. Well-mixed quasi-1-dimensional (Q1D) models of relative dispersion of two particles in the case of arbitrary Eulerian statistics were considered in our papers [108,114]. Recall that the Q1D model of relative dispersion is aimed at describing time evolution of the distance and longitudinal component of relative velocity between two particles. A 3-dimensional well-mixed model of relative dispersion consistent with arbitrary Eulerian statistics was proposed in [110]. LS models of relative dispersion based on the moments approximation approach have been proposed by Novikov [155], and have been developed in [77, 163, 164].

11.2 Basic assumptions We deal here with the process of relative dispersion of a pair of fluid particles in a stationary, spatially isotropic incompressible fully developed turbulent flow.

221

Section 11.2 Basic assumptions

We introduce the Eulerian velocity difference uE E .r/ D UE .x C r, t /  UE .x, t / considered at two fixed points separated by vector r.

11.2.1

Markov assumption

Let .r.t /, vE.t // be the Lagrangian variables for the separation vector and the relative velocity between two fluid particles. It is usually assumed that .r.t /, vE.t // is a 6D (continuous) Markov process (i. e., given the values of r.t / and vE.t / at time t , the values at time greater than t are independent of the values at times less than t). Under the Markov assumption for .r.t /, vE.t // the most general equation used to describe the time evolution of .r.t /, vE.t // is the Ito-type stochastic differential equation: dri D vi dt ,

dvi D ai .r, vE/ dt C bij .r, vE/ d Wj .t /,

i D 1, 2, 3.

(11.2)

The main problem here is the following: how can we determine the functions ai .r, vE/ and bij .r, vE/, (called the drift and diffusion terms, respectively) so that the model random process .11.2/ is in a sense close to the true process .11.1/? To this end, one uses two consistency principles: (i) consistency with Kolmogorov’s similarity theory and (ii) consistency with Thomson’s well-mixed condition.

11.2.2

Consistency with the second Kolmogorov similarity hypothesis

Given r, assume that there exists  such that    

r 2=3 . "N1=3

(11.3)

The consistency with the second Kolmogorov similarity hypothesis requires that N 1=2 ıij . bij .r, vE/ D .2 C0 "/

(11.4)

Here " is the mean rate of the dissipation of turbulence energy, and C0 is a universal constant in the Lagrangian structure function. Some comments: let us denote by V1 .t / D .V11 .t /, V12 .t /, V13 .t // and V2 .t / D .V21 .t /, V22 .t /, V23 .t // the Lagrangian velocity of the first and the second particles, respectively. Then,  vi .t /  vi .t C  /  vi .t / D V2i .t C  /  V2i .t /  .V1i .t C  /  V1i .t // D  V2i .t /   V1i .t /. In view of .11.3/, the quantities vE.t / D V2 .t /  V1 .t /,

 V2 .t /,

 V1 .t /

222

Chapter 11 Analysis of the relative dispersion of two particles

are approximately mutually independent. Therefore h vi .t / vj .t / j vE.t / D vE, r.t / D ri ® ¯ D h¹ V2i .t /   V1i .t /º  V2j .t /   V1j .t / j vE.t / D vE, r.t / D ri ' h V2i .t / V2j .t /i C h V1i .t / V1j .t /i ' 2 C0 "ı N ij  , which implies .11.4/.

11.2.3 Thomson’s well-mixed condition The following relation between the true Eulerian and Lagrangian p. d. f.’s are known [153]: Z v ; r, t / D pE .E

pL .r, vE; t , r0 / d r0 ,

where pE .E v ; r, t / D hı.E v  uE E .r//i, pL .r, vE; t , r0 / D hı.r  r.t //ı.E v  vE.t //i. The model is considered consistent with Novikov’s integral relation if its p. d. f. also satisfies such a relation. It is well known that this leads to Thomson’s well-mixed condition written in the form [237] vi

@ @2 pE @pE C .ai pE / D C0 "N . @ri @vi @vi @vi

It should be noted that all this does not define the model uniquely.

11.3 Well-mixed Lagrangian stochastic models Note that in the case of isotropic turbulence the structure of the drift term is defined by two scalar functions. Indeed, ai .r, vE/ D '.r, vk , v? /

ri C r

.r, vk , v? /

vi , v

where r D .ri ri /1=2 ,

vk D vi ri =r,

v? D .vi vi  vk2 /1=2 .

It should be noted that if we could find an additional relation between the functions ' and , then the well-mixed condition would provide a unique choice of the drift term. For instance, this is the case when '  0 (e. g., see the 1-particle model treated in [147], or  0 [114]. But generally, since such relations are not known,

223

Section 11.3 Well-mixed Lagrangian stochastic models

different approaches can be used to extract the unique model. We present below two such approaches. In the first one, the drift term is assumed to be quadratic in velocity, and the Eulerian p. d. f. pE is Gaussian ( [15, 238]. The second approach is based on a Markovian character of the evolution of the 2D process r.t /, ur .t / where r.t / is the distance between the two particles, and ur .t / is the longitudinal component of the relative velocity [108, 114].

11.3.1

Quadratic-form models

Following [15] let us assume that p. d. f. pE is Gaussian: pE .E u, r/ D

1=2 exp .2/3=2



1  ij ui uj , 2

where 1 ij .r/ D huE i .r/uEj .r/i,

 D det .ij /,

ik 1 kj D ıij ,

uE i .r/ D UE i .x C r/  UE i .x/. The drift term is constructed with the form ai D 'i C C0 "N

1 @pE , pE @ui

with 'i D i C ij k uj uk . As shown in [15], the well-mixed condition is satisfied for the following three cases: @j k 1 ij k D  1 , 2 il @rl @1 1 1 @ik il ij k D  1 D ,  kl 2 il @rj 2 @rj @1 @lj 1 1 il D ,  ij k D  1 jl 2 il @rk 2 @rk where

i D

1 ij

 1 @ C kkj C kj k .  2 @rj

Thus this approach gives a specific structure of the drift term, but it also does not provide a unique solution. Let us now consider the second approach.

224

Chapter 11 Analysis of the relative dispersion of two particles

11.3.2 Quasi-1-dimensional models The motion of two particles is described in the Lagrangian framework by the distance r.t / and the longitudinal velocity component defined here by ur .t /: r.t / D .ri .t /ri .t //1=2 ,

ur .t / D vi .t /ri .t /=r.t /.

Here vi .t / is the i -th component of the relative velocity vE.t /, and r.t / is the separation vector. Now we formulate the main assumption: Assume that .r.t /, ur .t // is a continuous 2D Markov process: dr D ur dt ,

dur D X.r, ur / dt C .2C0 "/ N 1=2 d W .t /.

(11.5)

where X.r, ur / is an unknown drift term. In the case of quasi-1-dimensional model the well-mixed condition reads [114] k

v

k

k

@.X.r, v/pE / @2 pE @r 2 pE .v, r/ N 2 , C r2 D C0 "r @r @v @v 2

(11.6)

k

where pE .v, r/ D hı.v  uE r /i is the p. d. f. of uE r , and the longitudinal component of the Eulerian velocity difference uE E .r/ D UE .x C r/  UE .x/, i. e., uE r D .UE i .x C r/  UE i .x//

ri . r

k

Assuming XpE jjvj!1 D 0, it is easy get X.r, v/ D C0 "N

@ 1 k ln pE .v, r/  k @v pE .v, r/

Z

v

1

v0 @  2 k 0  0 r pE .v , r/ dv . r 2 @r

In the inertial subrange (  r  L) this expression can be considerably simplified. In the inertial subrange the unique external parameter is "N, so it is possible to turn to a nondimensional density fE ./: k

pE .v, r/ D

1 v fE . /. 1=3 .N"r/ .N"r/1=3

Consequently,  "N2=3 Q v X.r, v/ D 1=3 X , r .N"r/1=3 2 d ln fE Q C  X./ D C0 d 3

7 3

R

1 

0f

E .

fE ./

0/ d  0

.

225

Section 11.3 Well-mixed Lagrangian stochastic models

It is convenient to use the equation in the nondimensional form r .t / D ur .t /=.N"r/1=3 :  1=3  1=6 p "N "N 1=3 dr D .N"r/ r dt , d r D a0 .r /dt C 2 2C0 d W .t /, 2 r r where 2 d ln fE Q a0 ./ D X./  D C0  3 d In the case of Gaussian p. d. f. fE fE ./ D

7 3

R

1 

0f

E .

fE ./

0/ d  0

.

 1 2 exp  , 2C .2 C /1=2

the coefficient a0 has the most simple form: a0 ./ D 

7 C0 C C. C 3

Here C is the Kolmogorov universal constant defined by hu2E r i D C.N" r/2=3 .

11.3.3

3-dimensional extension of Q1D models

Assume that the turbulence is isotropic and stationary. Let us consider a 3D model of relative dispersion in the subrange   r: dri D vi dt ,

dvi D ai .r, vE/ dt C .C0 "N/1=2 d Wi .t /,

where ai .r, vE/ D '.r, vk , v? /

ri C r

.r, vk , v? /

i D 1, 2, 3, vi v

with unknown ' and . We derive from it the Q1D model [110]:  u2? ai ri C dt C .C0 "/ N 1=2 d W .t /, dr D ur dt , dur D r r where u2? .t / D u2 .t / u2r .t /. Assuming that Q1D process .r.t /, vr .t // is Markovian, we can write v2 v2 vk ai ri C ? D '.r, vk , v? / C .r, vk , v? / C ? D X.r, vk /. r r v r From the last relation and the 3D well-mixed condition it follows that vk v v @ ln pE .vk , v? ; r/ C v? @v? r ³ Z v? ² vk @ 2 v @ 0 0 0  2 .XpE / C 2 r pE .vk , v? ; r/ v? dv? , v? pE .vk , v? ; r/ 0 @vk r @r

.r, vk , v? / D C0 "N

226

Chapter 11 Analysis of the relative dispersion of two particles

2 v? vk  .r, vk , v? /. r v This yields in the case of Gaussian p. d. f. pE , for the inertial subrange   r  L

'.r, vk , v? / D X.r, vk / 

ai .r, vE/ D

"N2=3 h 7 v 2 i ri C 0 vk  C  4C .N"r/1=3 .N"r/2=3 r r 1=3 3 C 0 i vi "N2=3 h 4 vk  , C 1=3 3 .N"r/1=3 C 0 .N"r/1=3 r

where C 0 D 43 C .

11.4 Stochastic Lagrangian models based on the moments approximation method The evaluation of the p. d. f. pE .E v ; r/ is a very difficult problem, because it needs to construct a family of solutions to Navier–Stokes equations. Therefore, in practice one uses the method of moments: one constructs an approximation to the Eulerian p. d. f. under the condition that its first several moments coincide with those of the true velocity moments. The true moments can be found via DNS method solving the Navier–Stokes equation, or extracted from experiments.

11.4.1 Moments approximation conditions In a more general case, when the intermittency is taken into account, the model has the form dri D vi dt ,

dvi D ai .r, vE/ dt C bij .r, vE/ d Wj .t /,

i D 1, 2, 3.

The well-mixed condition reads in this case (e. g., see [155, 237]) vi

@ 1 @2 mij pE @pE C .ai pE / D , @ri @vi 2 @vi @vj

(11.7)

where mij D bik bj k . Multiplying this equation by vj and integrating over vE, we get [164] @ huE i .r/uEj .r/i D haj .r, uE E .r//i, @ri i. e., (due to incompressibility) haj .r, uE E .r//i D 0,

j D 1, 2, 3.

227

Section 11.4 Models based on the moments approximation

Multiplying .11.7/ by vj vk and integrating over vE yields @ huE i .r/uEj .r/uE k .r/i @ri D haj .r, uE E .r// uE k .r/ C ak .r, uE E .r// uEj .r/i C hmj k .r, uE E .r//i for j , k D 1, 2, 3. Kolmogorov’s relation for third-order moments in the inertial subrange (e. g., see [146, 155]) is huE i .r/uEj .r/uE k .r/i D 

4 "N.ri ıj k C rj ıik C rk ıij /, 15

  r  L.

Hence

4 @ huE i .r/uEj .r/uE k .r/i D  "Nıj k , .  r  L/. @ri 3 Therefore, in the inertial subrange the moments approximation conditions have the form 4 haj .r, uE E .r// uE k .r/ C ak .r, uE E .r// uEj .r/i C hmj k .r, uE E .r//i D  "Nıj k , 3 j , k D 1, 2, 3; .  r  L/. (11.8) haj .r, uE E .r//i D 0,

11.4.2

Realizability of LS models based on the moments approximation method

Here we have to analyse whether or not the scheme presented in the previous subsection can indeed be realized. The input parameters of the moments approximation method are huE i .r/uEj .r/i,

huE i .r/uEj .r/uE k .r/i,

i , j , k D 1, 2, 3.

Let us assume that the functions ai and bij satisfying the moments approximation conditions .11.8/ are found. v , r/ to equation Now we have to check whether there exists a positive solution pE .E .11.7/ which satisfies the conditions Z Z pE .E v , r/ d vE D 1, vi pE .E v , r/ d vE D 0, IR3

Z

IR3

IR3

vi vj pE .E v , r/ d vE D huE i .r/uEj .r/i,

Z

IR3

vi vj vk pE .E v , r/ d vE D huE i .r/uEj .r/uE k .r/i,

i , j , k D 1, 2, 3.

228

Chapter 11 Analysis of the relative dispersion of two particles

Example ([164]). .N"r/1=3 ai D k .vi C vj nj ni /, ni D ri =r, r ± ° mij D "N ˇıij C .˛  ˇ/ni nj , .˛  0, ˇ  0/,

(11.9) (11.10)

where ˛, ˇ, and k are dimensionless coefficients of the model (˛  0, ˇ  0, k  0), and C is Kolmogorov’s constant in the law of two-thirds (C ' 2). The moments approximation conditions .11.8/ imply the following relations between these coefficients [164]: 8C k D 3ˇ C 4,

.6  2/C k D .3˛  ˇ/.

From ˛  0, ˇ  0 it follows that C k  0.5,

.9 C 1/C k  2.

Thus for any and k  0 satisfying these inequalities the model .11.9/–.11.10/ satisfies the moments approximation conditions .11.8/. However as shown in [164], the realizability conditions are satisfied only for a special subrange of the pairs . , k/. Some examples of such pairs are: (1/3,3), (1,2.5), (3,1.3), (5,0.9). A generalization of the model .11.9/–.11.10/ is given in [163], where the drift term is defined by  k (11.11) ai D  .N"r/1=3 .vi C vj nj ni / C  vj nj vQ i , vQ i D vi  vj nj ni r and with diffusion term .11.10/. In absence of intermittency, the moments approximation conditions .11.8/ imply [163] 4 ˛ D  C 2k. C 1/, 3

4 8 ˇ D  C k.5C  /. 3 15

Realizability conditions imply that the resting parameters k, and  are not completely free. For example, as shown in [163], in the case of isotropic forcing .˛ D ˇ/ which implies D 1=3  4=15C , and assuming  D 0, 0.25, 1, the corresponding values of parameter k, for which the realizability conditions are satisfied, are 3.8, 4.2, and 6, respectively. The examples cited will be used below in the next section when comparing different models of relative dispersion in the inertial subrange (see Table 11.1).

229

Section 11.5 Comparison of different models of relative dispersion

11.5

Comparison of different models of relative dispersion for the inertial subrange of a fully developed turbulence

11.5.1

Q1D quadratic-form model of Borgas and Yeung

The exact Eulerian transport equation for the Eulerian velocity p. d. f. pE .E v , r/ is (e. g., see [77]) @ @pE vi C .hai jE v , ripE / D 0, @ri @vi i is the relative acceleration between two fluid particles, and hai jE v , ri where ai .t / D du dt is its conditionally averaged value. The Q1D analog of this equation is the following k exact transport equation for pE , e. g., see [17]):

k

@ ur @r 2 pE k .har jur , ripE / D 0, C 2 r @r @ur where ar .t / D

(11.12)

dur .t / ai .t /ri .t / vi .t /vi .t /  u2r .t / D C dt r.t / r

is the longitudinal component of the relative acceleration, and har jur , ri is the conditional acceleration. From .11.6/ and .11.12/ it follows that k

1 @pE .ur , r/ C har jur , ri. X.r, ur / D C0 "N k @ur p

(11.13)

E

The quadratic-form assumption for the conditional acceleration is har jur , ri D ˛.r/ C ˇ.r/ur C .r/u2r D

2 .˛0 C ˇ0  C 0  2 / , r

(11.14)

where  2 D hu2E r i,  D ur = . In order to satisfy the moment constraints hu2E r i D  2 ,

hu3E r i D S 3 ,

hu4E r i D F  4

the coefficients ˛, ˇ, and should be chosen from the following equations [17]: 1 d.r 2  2 / D ˛ C  2 , r 2 dr 1 d.r 2 S 3 / D 2ˇ 2 C 2 S 3 , r2 dr 1 d.r 2 F  4 / D 3˛ 2 C 3ˇS 3 C 3 F  4 , r2 dr

230

Chapter 11 Analysis of the relative dispersion of two particles

which can be solved for parameters ˛, ˇ and to give ˛ D ˛0

2 , r

 ˇ D ˇ0 , r

1 D 0 , r

(11.15)

where the dimensionless parameters ˛0 , ˇ0 , and 0 are ˛0 D  0  2

0 D

0

3 ˇ0 D  0  S 2

C 2,

 6  3S 2 C 2F C

0 .6

0

CS C

r dS , 2 dr

0

C 92 S 2  4F /  32 S r dS C r dF dr dr

3.F  S 2  1/

D r

d ln  , dr

 .

(11.16)

In the inertial subrange   r  L we have  2 .r/ D C.N"r/2=3 , S D const, F D const , due to Kolmogorov’s second similarity hypothesis (e. g., see [146]) which yields 0 D 1=3. Therefore .11.16/ yields 0 D

 32 S 2  83 , F  S2  1

10 9F

˛0 D

8  0 , 3

3 ˇ0 D S  0 . 2

In the inertial subrange there exists a dimensionless p. d. f. f ./ such that  1 ur k f . pE .ur , r/ D  .r/  .r/ Then in this case equation .11.12/ can be rewritten as the following ODE: .b0 C b1  C b2  2 /

df ./ C .c0 C c1 /f ./ D 0, d

where b0 , b1 , b2 , c0 , c1 are the dimensionless constants b0 D ˛0 ,

1 b2 D 0  , 3 1 c1 D 2 C 2 0  . 3

b1 D ˇ0 ,

c0 D b1 ,

From .11.13/, .11.14/, and .11.17/ we get X.r, ur / D where

"N2=3 Q X r 1=3



ur ,  .r/

 .r/ D

p

C .N"r/1=3 ,

C0 c0 C c1  XQ ./ D  p C C.˛0 C ˇ0  C 0  2 /. C b0 C b1  C b2  2

(11.17)

231

Section 11.5 Comparison of different models of relative dispersion

Table 11.1. The universal constant g in the Richardson law hr 2 .t /i D g"t 3 calculated by different well-mixed Lagrangian stochastic models. Well-mixed models

C0 D 4

C0 D 5

C0 D 6

C0 D 7

C0 D 10

Borgas and Sawford .1994/ Thomson (1990) Q1D, p. d. f. of Borgas and Yeung (1998) Q1D, bi-Gaussian p. d. f. 1 D 2 . Q1D, bi-Gaussian p. d. f.

1 =1 D  2 =2 . Q1D, Gaussian p. d. f.

0.75

0.7

0.6

0.5

0.3

1.8 4.8

1.4 2.6

1.1 1.7

0.85 1.1

0.45 0.45

4.4

2.55

1.67

1.15

0.47

8.25

4.

2.27

1.4

0.51

7.

4.7

2.8

1.9

0.75

11.5.2

Comparison of different models in the inertial subrange

In this subsection we study the process of the relative dispersion of two particles in the inertial subrange (ISR)   r  L. We first consider Q1D models which are determined by the Eulerian p. d. f. fE ./ of the dimensionless velocity difference r D uE r =.N"r/1=3 . Suppose ² ² ³ ³ p .  1 /2 .  2 /2 1p fE ./ D p exp  p exp  C , 212 222 21 22 where the unknown parameters p, 1 , 2 , 1 , 2 should be chosen to fit the first four moments of r : hr i D p 1 C .1  p/ 2 D 0, hr2 i D p. 21 C 12 / C .1  p/. 22 C 22 / D C , 4 hr3 i D p. 31 C 3 1 12 / C .1  p/. 32 C 3 2 22 / D  , 5 hr4 i D p. 41 C 6 21 12 C 314 / C .1  p/. 42 C 6 22 22 C 324 / D 3.4 C 2 . Thus we have four equations for five unknown parameters p, 1 , 2 , 1 , 2 . To find these parameters, we need an additional (closure) assumption. Further we will use two kinds of closure assumptions: (i) 1 D 2 , and (ii) 1 =1 D  2 =2 . In Table 11.1 we present the Richardson constant g obtained by different models. In the models [163, 164], the isotropic forcing was considered (˛ D ˇ; see Section 4.2)

232

Chapter 11 Analysis of the relative dispersion of two particles

Table 11.2. The universal constant g evaluated by different moments approximation models. Model

realizability

Pedrizzetti & Novikov Pedrizzetti Heppe

+ + + unknown unknown

g 0.50 0.35 0.20 0.44 0.31

.k D 3.8, C0 D 9.47/ . D 0.25, k D 4.2, C0 D 10.25/ . D 1, k D 6, C0 D 13.73/ .C0 D 5.3/ .C0 D 9.47/

Table 11.3. The universal constant g evaluated by other methods. Method LHDI, Kraichnan 1966 [101] Modified LHDI, Lundgren 1981 [135] EDQNM, Larcheveque & Lesieur 1981 [125] Thomson’s corrected EDQNM, Thomson 1996 [239] Effective Hamiltonian method, Nakao 1991 [149]

g 2.42 3.0 3.5 1.4 3.5

11.6 Comparison of different Q1D models of relative dispersion for modestly large Reynolds number turbulence (Re ' 240) The models discussed in the previous section were developed for the case of a turbulence with a reach inertial subrange. In this case the DNS methods cannot be used, since today the DNS data is available around Re ' 240. Therefore, to make the validation through comparison with the DNS method, we can do this only for models with modestly large Reynolds-number turbulence.

11.6.1 Parametrization of Eulerian statistics Here we turn to a turbulent flow with L= ' 500. In this case the scales go from the viscous ones, pass through a transitional region, and goes on to the external interval. But it should be noted that the transitional range can be considered to be an analog of the inertial subrange only in terms of the Eulerian statistical characteristics. For the Lagrangian statistics however this interval does not show the inertial subrange behavior. Thus we have to make the assumption that r.t /, vE.t / is a Markov random process governed by .11.5/ with the value C0 depending on the distance r such that C0 tends to a constant value as r= is increases.

233

Section 11.6 Comparison of different Q1D models of relative dispersion

Dimensional arguments show that in the case of stationary and isotropic turbulence k

pE .v, r/ D

1 v fE . ; r/,  .r/  .r/

where  .r/ D hu2E r i1=2 is r. m. s of the longitudinal component of the Eulerian velocity difference uE E .r/ D UE .x C r/  UE .x/: uE r D .UE i .x C r/  UE i .x//

ri , r

and fE .; r/ is a dimensionless p. d. f. of dimensionless velocity r D uE r = .r/. It follows from .11.13/ that X.r, v/ D

v "N2=3 Q /, X.r, 1=3  .r/ r

N @ ln fE Q / D C0 .r/ "r X.r,   3 @

R 0 .r/

2

 .2 

0/

1 

0f

E .

0 ; r/ d  0

fE .; r/

,

where

d ln  . dr If we pass to the dimensionless velocity r .t / D ur .t /= .r.t //, then the Q1D model reads ± .C .r/N"/1=2  .r/ ° Q 0 X.r, r / C 0 .r/r2 C d W .t /. dr D  .r/r dt , d r D r  .r/ (11.18) 0

D

0 .r/

D r

Now we have to specify the Eulerian velocity fE .; r/ and the function  .r/. Concerning the density fE .; r/, we assume that the first four moments are given in [17]: hr i D 0,

hr2 i D 1,

h 3 i D

hu3E r i hu2E r i

, 3=2

h 4 i D

hu4E r i , hu2E r i2

where 

2=3  1=3 r2 r2 D , !2 2 C r 2 ƒ2 L2 C r 2  4  r 3  L2 L2 hu3E r i D  13 , L !3 2 C r 2 ƒ3 L2 C r 2  4=3  2=3 r2 r2 . hu4E r i D 4k 14 !4 2 C r 2 ƒ4 L2 C r 2

u2E r .r/

212

234

Chapter 11 Analysis of the relative dispersion of two particles

Here 1 D ŒUE i UE i 1=2 is the one-point root-mean-square (rms) velocity fluctuations, L D 13 =N", the dimensionless constants  , k , !2 , !3 , !4 , ƒ2 , ƒ3 , ƒ4 in this parametrization are  3  3=2 2 4 k , ƒ3 D 1., ƒ4 D ƒ2 , k D 3.,  D  .ƒ3 /4 , ƒ2 D C Ki 5  3=4 1=2 164.32 4 !2 ƒ2 Ki !2 D p , !3 D  , ! D ! , 4 2 5 23=2 S0 K0 ƒ2 K0 D 7.5,

S0 D 0.5,

Ki D 3.4 .

11.6.2 Bi-Gaussian p. d. f. Here we compare the Lagrangian statistical characteristics obtained by the model .11.18/ with the results obtained by DNS. The p. d. f. fE .; r/ is chosen as the following bi-Gaussian density with unknown parameters p, 1 , 2 , 1 , 2 : ² ² ³ ³ p .  1 /2 .  2 /2 1p fE ., r/ D p exp  exp  Cp , 212 222 21 22 p 1 C .1  p/ 2 D 0,

p. 21 C 12 / C .1  p/. 22 C 22 / D 1,

p. 31 C 3 1 12 / C .1  p/. 32 C 3 2 22 / D S , p. 41 C 6 21 12 C 314 / C .1  p/. 42 C 6 22 22 C 324 / D F , hu3 i

hu4 i

where S D S.r/ D hu2 Eri3=2 and F D F .r/ D hu2Eri2 are the skewness and the Er Er flatness of uE r , respectively. To determine uniquely the five unknown parameters, we use the closure assumption [125] which has proven to be reasonable:

2

1 D D m, 1 2

(11.19)

where m D m.r/ is the solution to the equation  .1 C m2 /3 S 2 3 C 6m2 C m4 F D 1C . .3 C m2 /2 m2 .1 C m2 /2 The assumption .11.19/ allows us to obtain the unknown parameters 1 and 2 explicitly:   1=2 1=2 1p p , 2 D , 1 D p.1 C m2 / .1  p/.1 C m2 /

235

Section 11.6 Comparison of different Q1D models of relative dispersion hr 2 i1=2 = 1,000

100

10

1

10 t =

100

Figure 11.1. The dimensionless rms of the relative separation; Data obtained by the Lagrangian stochastic model: r0 D 16 (lower solid curve) and r0 D 64 (upper solid curve). The relevant DNS data are shown as the dotted curves: r0 D 16 (lower dotted curve) and r0 D 64 (upper dotted curve).

where

 1=2  1 a pD 1 , 2 4Ca

aD

.1 C m2 /3 S 2 . .3 C m2 /2 m2

In Figure 11.1, the dimensionless rms of the relative separation as a function of dimensionless time is presented. Calculations by the model .11.18/ were carried out for two initial separations: r0 D 16 (lower solid curve) and r0 D 64 (upper solid curve). The relevant DNS data are shown as dotted curves. In Figure 11.2 the skewness factor is shown as a function of dimensionless time. In this curve, the upper solid lines correspond to the initial separation r0 D 16, and the lower solid lines to r0 D 64. The DNS data are also shown as dotted curves, where r0 D 16 and r0 D 64 correspond to the upper and lower lines, respectively. As the results of Figure 11.1 show, the model describes the rms of the separation well for dimensionalless times from 0 to 10. There is some discrepancy between the model and DNS data for larger times. As to the skewness and flatness factors, we can only state a qualitative agreement with the DNS data: the model overestimates these factors, compared to the DNS results, in the time interval intermediate between the viscous and external ranges. It should be noted that our calculations with the bi-Gaussian p. d. f. with other types of closure assumptions qualitatively show the same picture. We conclude thus that the bi-Gaussian p. d. f. is not the right choice. Therefore we try another class of densities,

236

Chapter 11 Analysis of the relative dispersion of two particles 2 skewness 1.5 1 0.5 0 0.5

1

10 t =

100

Figure 11.2. The skewness factor of the relative separation; Data obtained by the Lagrangian stochastic model: r0 D 64 (lower solid curve) and r0 D 16 (upper solid curve). The relevant DNS data are shown as the dotted curves: r0 D 64 (lower dotted curve) and r0 D 16 (upper dotted curve).

based on the quadratic-form approximation of the conditional acceleration. These densities have a different tail behavior, which might have a large impact on the skewness and flatness factors. Such indications were reported by M. Borgas [14].

11.6.3 Q1D quadratic-form model Let us show that the realizability condition will be satisfied if there exists a positive k solution pE to equation .11.12/ with the quadratic-form of the conditional acceleration .11.14/ which is a p. d. f. of the variable uE r : Z 1 k pE .ur ; r/ dur D 1, 8r  0, (11.20) 1

and there exists a point r0 such that uE r .r0 / D 0,

u2E r .r0 / D  2 .r0 /,

u3E r .r0 / D .r0 /,

u4E r .r0 / D #.r0 /. (11.21)

Here and below we denote by an overbar the average over this p. d. f.. In order to prove this assertion let us integrate equation .11.12/ on ur to get r 2 uE r D const which yields uE r .r/ D 0, since uE r .r0 / D 0. Further multiplying equation .11.12/ by ur and then integrating over ur , we get 1 dr 2 u2E r D ˛ C u2E r . r 2 dr

237

Section 11.6 Comparison of different Q1D models of relative dispersion

This yields u2E r .r/ D  2 .r/ since the functions u2E r and  2 .r/ satisfy the same equations and the common initial conditions at r D r0 (see .11.21/). Multiplying equation .11.12/ by u2r and u3r and integrating over ur we find that the functions u3r and u4E r satisfy the same equations as the functions .r/ and #.r/, respectively. From these and the conditions .11.21/ it follows that u3r .r/ D .r/,

u4r .r/ D #.r/,

i. e., the realizability condition is established. Remark 11.1. If we assume that for some ˛, r1 , and r2 the conditions hu2E r i D const, r˛

S.r/ D const,

F .r/ D const,

r1  r  r2 . (11.22)

are valid, then from .11.16/ and .11.22/ it follows that the coefficients b0 , b1 , b2 , c0 , and c1 are constants, say, bQ0 , bQ1 , bQ2 , cQ0 , cQ1 in the subrange r1  r  r2 . For example in the far-viscous subrange r  , in the inertial subrange   r  L, and in the external subrange r L these conditions are fulfilled. This property can be used to define the boundary conditions to the transport equation for the function f ., r/: r

@f ., r/ @f ., r/ C .b0 C b1  C b2  2 / C .c0 C c1 /f ., r/ D 0, @r @

(11.23)

where b0 , b1 , b2 , c0 , c1 are dimensionless functions depending on r: b0 .r/ D ˛0 .r/,

b1 .r/ D ˇ0 .r/,

c0 .r/ D b1 .r/, and ˛0 .r/, ˇ0 .r/, 0 .r/ and solution to the ODE

b2 .r/ D 0 .r/ C

c1 .r/ D 2 C 2 0 .r/ C

0 .r/

0 .r/,

0 .r/,

are determined in .11.16/. Indeed, let f0 ./ be a

df0 ./ C .cQ0 C cQ1 /f0 ./ D 0, .bQ0 C bQ1  C bQ2  2 / d satisfying the condition

Z

1 1

f0 ./ d  D 1.

We can consider the p. d. f. f0 ./ as an approximation to the p. d. f. f ., r/ in the subrange r1  r  r2 . Choosing r0 in the interval r1  r0  r2 , we can put the following boundary condition for equation .11.23/: f ., r D r0 / D f0 ./.

(11.24)

This brief remark is just to recall that there are many possible choices of the Eulerian p. d. f..

Chapter 12

Evaluation of mean concentration and fluxes in turbulent flows by Lagrangian stochastic models Forward and backward stochastic Lagrangian trajectory simulation methods for calculation of the mean concentration of scalars and their fluxes for sources arbitrarily distributed in space and time are constructed and justified. Generally, absorption of scalars by medium is taken into account. A special case of the source structure, when the scalar is generated by a plane source, say, located close to the ground, is treated. This practically interesting particular case is known in the literature as the footprint problem.

12.1 Introduction The turbulent dispersion of particles in the framework of statistical fluid mechanics is described as the transport of particles in a random velocity field (e. g., [146]). In particular, the concentration of scalars and their fluxes are random fields. There are mainly two different approaches for calculation of the mean values of these fields: (i) the conventional deterministic methods, based on the semiempirical turbulent diffusion equation and closure assumptions (e. g., see [79,162,225]), and (ii) the stochastic approach, which utilizes trajectory simulations (e. g., see [62, 102, 113, 191, 237, 244, 256]). The deterministic approach directly deals with the equation governing the mean concentration, and relies on the Boussinesq hypothesis, whose applicability is restricted (e. g., see [19,175]). For instance, this hypothesis cannot be true if the concentration is calculated close to the sources [19, 146]. More generally, the high-order closure methods are developed, but a different closure hypothesis also should be made (see e. g., [95, 146]). The stochastic approach based on the modeling of stochastic Lagrangian trajectories in principle does not require a closure hypothesis. Two main issues in this approach are (i) the development of adequate Lagrangian stochastic models governed by generalized Langevin-type equations and (ii) construction of Monte Carlo random estimators for the evaluation of desired statistical characteristics (for instance, the mean concentration, the mean height of a cloud of particles, etc.). It should be noted that in the Monte Carlo methods, when using the random estimators the results are obtained with statistical errors. Recall that a random variable  is said to be a Monte Carlo estimator for a quantity a if the mathematical expectation of  is equal to a: IE D a. If 1 , 2 , : : : , N are N -independent samples of PN the random variable , then the average SN D N1 iD1 i almost surely tends to a

239

Section 12.2 Formulation of the problem

(i. e., with probability 1) as N tends to infinity, and the error in using SN to approximately a D IE (for sufficiently large N ) is proportional to the standard deviation of . As N increases, this statistical error decreases as N 1=2 . The p well-known “law of three sigmas” gives the rate of convergence: IP.jSN  aj < 3 = N /  0.997. Here  D .IE 2  IE2 /1=2 is the standard deviation of . The larger N is, the closer the distribution of SN to the Gaussian one, and the better the approximation. Issue (i) attracted attention in many publications (e. g., see [115,178,200,255,256]). In this chapter we concentrate on issue (ii). It should be noted that this field is not well developed, and we can give only a few references [57, 111, 215, 237]. In this chapter we treat simulation methods based on the forward and backward Lagrangian trajectories. The general principle is quite clear: one uses the backward trajectories originating at the detector if it is a point detector in space (or the detector occupies a small volume); the forward trajectories are used if the detector is quite extended in space. The chapter is organized as follows. In Section 12.2 one relates the calculation of the mean concentration and its flux with the averages over Lagrangian trajectories governed by generalized Langevin-type equations. The forward and backward estimators are presented in Section 12.3. Applications of these estimators to the footprint problem are given in Section 12.4. Some technical details are included in Appendices A–C at the end of the chapter.

12.2

Formulation of the problem

Let us assume that a passive but generally nonconservative scalar is dispersed by a turbulent velocity field uE .x, t / in the half-space D D ¹x D .x1 , x2 , x3 / : x3  0º, for example in the surface layer of the atmosphere. Throughout this chapter the following notation of spatial and velocity co-ordinates is used: x D .x1 , x2 , x3 / D .x, y, z/ and uE D .u1 , u2 , u3 / D .u, v, w/; and analogously X D .X1 , X2 , X3 / D .X , Y , Z/ and V D .V1 , V2 , V3 / D .U , V , W / for the Lagrangian coordinates. The passive scalar is assumed to be uninertial, i. e., it follows the streamlines of the flow. The evolution of scalar concentration field from a source of intensity q.x, t / (the amount of emitted scalar per unit volume in a unit time interval at the phase point .x, t /) is controlled by the turbulent transport and absorption by a medium: @c @c.x, t / C .x, t /c.x, t / D q.x, t /, t > 0; C ui .x, t / @t @xi

c.x, 0/ D q0 .x/,

where .x, t / (  0) denotes the coefficient of absorption, the initial spatial distribution of concentration is given by q0 .x/, and the molecular diffusion is neglected. Here and in the following the summation convention is assumed over repeated indices.

240

Chapter 12 Evaluation of mean concentration and fluxes

The turbulent velocity field uE .x, t / is assumed to be an incompressible 3-dimensional (3D) random field. Accordingly the concentration c.x, t / is also a scalar random field. We consider the simplest statistical characteristics of this field, the mean concentration hc.x, t /i, the mean flux of scalar concentration hui .x, t /c.x, t /i, and the spatial-temporal average of these statistical characteristics. Here and below the angle brackets denote the average over samples of turbulent velocity fluctuations. The above-mentioned means are calculated by simulation of the Lagrangian trajectories X.t / D X.t ; x0 , t0 /, t  t0 , determined by dXi .t / D ui .X.t /, t / D Vi .t /, dt

X.t0 / D x0 ,

where V.t / D V.t ; x0 , t0 / is the Lagrangian velocity. The instantaneous concentration can be expressed as (see Appendix A at the end of the chapter) Z t Z dt0 d x0 .t ; x0 , t0 /q.x0 , t0 /ı.x  X.t ; x0 , t0 // c.x, t / D 0 D Z C d x0 .t ; x0 , 0/q0 .x0 /ı.x  X.t ; x0 , 0//, (12.1) D

where ı./ is the Dirac delta function, and .t / D .t ; x0 , t0 / is defined by d .t / C .X.t ; x0 , t0 /, t / .t / D 0, dt

.t0 / D 1.

(12.2)

The expression for instantaneous concentration can be rewritten as Z Z 1 Z t Z c.x, t / D d uE d

dt0 d x0 Q.x0 , t0 / IR3

0

D

0

ı.x  X.t ; x0 , t0 //ı.E u  V.t ; x0 , t0 //ı.  .t ; x0 , t0 //, where Q.x0 , t0 / D q.x0 , t0 /Cq0 .x0 /ı.t0 /. Averaging the last equation yields the mean concentration Z 1 Z t Z Z d uE d

dt0 d x0 Q.x0 , t0 /pL .x, uE , , t ; x0 , t0 /, (12.3) hc.x, t /i D IR3

0

0

D

where u  V.t ; x0 , t0 //ı.  .t ; x0 , t0 //i pL .x, uE , , t ; x0 , t0 / D hı.x  X.t ; x0 , t0 //ı.E is the joint probability density function (p. d. f.) of Lagrangian characteristics X.t ; x0 , t0 /, V.t ; x0 , t0 /, and .t ; x0 , t0 /.

241

Section 12.2 Formulation of the problem

Analogously, the mean flux of concentration can be represented as Z Z 1 Z t Z d uE d

dt0 hui .x, t /c.x, t /i D IR3

0

0

D

d x0 ui Q.x0 , t0 /pL .x, uE , , t ; x0 , t0 /,

i D 1, 2, 3.

(12.4)

For convenience, the mean characteristics (12.3) and (12.4) will be written in the general form hg.E u.x, t //c.x, t /i Z 1 Z t Z Z d uE d

dt0 d x0 g.E u/ Q.x0 , t0 /pL .x, uE , , t ; x0 , t0 /, (12.5) D IR3

0

D

0

where g.E u/ equals 1 and ui for the mean concentration and fluxes, respectively. Our problem can be formulated as follows. It is necessary to represent the integral (12.5) as an expectation of a random estimator defined on Lagrangian trajectories. But since the exact form of pL .x, uE , , t ; x0 , t0 / is not known, we have to use some approximation, usually taken as a p. d. f. of the solution to the following generalized Langevin-type equation (e. g., see [237]): d X.t / D V.t /dt , d V.t / D a.t , X.t /, V.t //dt C

p

C0 ".X.t N /, t / d W.t /,

(12.6)

where C0 is the universal Kolmogorov constant (C0  46), ".x, N t / is the mean dissipation rate of the kinetic energy of turbulence, and W.t / D .W1 .t /, W2 .t /, W3 .t // is the standard 3D Wiener process. The function a is to be specified in each specific situation (e. g., [56,133,237,256]). We mention only that in all these models Thomson’s well-mixed condition should be satisfied [237]. We will deal in this chapter with two different types of random estimators, namely, with forward estimators, which are defined on forward Lagrangian trajectories. which emanate from the source and move toward the detector, and with backward estimators, which are defined on backward trajectories starting at the detector and moving toward the source. More exactly, the solution to (12.6) with the initial conditions X.t0 / D x0 ,

V.t0 / D uE 0

is called the forward Lagrangian trajectory. We denote it by Xxt 0 ,uE 0 ,t0 and Vxt 0 ,uE 0 ,t0 . Then the true Lagrangian trajectory X.t ; x0 , t0 /, V.t ; x0 , t0 / can be approximated by the model trajectory Xxt 0 ,uE 0 ,t0 and Vxt 0 ,uE 0 ,t0 with the random initial velocity uE 0 chosen according to the Eulerian p. d. f. pE , which is defined by pE .E u; x0 , t0 / D hı.E u uE .x0 , t0 //i.

242

Chapter 12 Evaluation of mean concentration and fluxes

Let pL .x, uE , , t ; x0 , uE 0 , t0 / be the conditional p. d. f. under the condition that V.t ; x0 , t0 / D u0 : pL .x, uE , , t ; x0 , uE 0 , t0 / ˇ ˛ ˝ D ı.x  X.t ; x0 , t0 //ı.E u  V.t ; x0 , t0 //ı.  .t ; x0 , t0 //ˇ V.t ; x0 , t0 / D u0 . By the theorem on conditional probability we get Z pL .x, uE , , t ; x0 , t0 / D d uE 0 pE .E u0 ; x0 , t0 /pL .x, uE , , t ; x0 , uE 0 , t0 /. IR3

Let us introduce the model conditional p. d. f. as f

pL .x, uE , , t ; x0 , uE 0 , t0 /

® ¯ u  Vxt 0 ,uE 0 ,t0 /ı.  xt 0 ,uE 0 ,t0 / , (12.7) D IEx0 ,uE 0 ,t0 ı.x  Xxt 0 ,uE 0 ,t0 /ı.E

where .t / D xt 0 ,uE 0 ,t0 is defined by d .t / C .Xxt 0 ,uE 0 ,t0 , t / .t / D 0, dt

.t0 / D 1.

Here IEx0 ,uE 0 ,t0 means the expectation over samples of stochastic processes Xxt 0 ,uE 0 ,t0 ,

Vxt 0 ,uE 0 ,t0 , and xt 0 ,uE 0 ,t0 , starting at time t D t0 from the point x0 , uE 0 , 1. f

Taking the model transition density pL .x, uE , , t ; x0 , uE 0 , t0 / as an approximation to pL .x, uE , , t ; x0 , uE 0 , t0 /, the true Lagrangian p. d. f. pL .x, uE , , t ; x0 , t0 / is approximated as Z f d uE 0 pE .E u0 ; x0 , t0 /pL .x, uE , , t ; x0 , uE 0 , t0 /. (12.8) pL .x, uE , , t ; x0 , t0 /  IR3

Substituting the approximation (12.8) into the integral (12.5), we come to the approximate equality Z 1 Z t Z Z d uE d

dt0 d x0 g.E u/ Q.x0 , t0 / hg.E u.x, t //c.x, t /i D IR3 D 0 0 Z f d uE 0 pE .E u0 ; x0 , t0 /pL .x, uE , , t ; x0 , uE 0 , t0 /. (12.9) IR3

E O 0/ D X O x,u,t The backward Lagrangian trajectory, denoted in the following by X.t t0 ,

E O 0/ D V O x,u,t V.t t0 , t0  t , is defined as the solution to (e.g, [57, 237])

O 0 /dt0 , O 0 / D V.t d X.t O 0 /, V.t O 0 //dt0 C O 0 / D aO .t0 , X.t d V.t

q

O 0 /, t0 / d W.t0 /, C0 "N.X.t

(12.10)

243

Section 12.3 Monte Carlo estimators for the mean concentration and fluxes

O / D x, V.t O / D uE , where the drift term aO D .aO 1 , aO 2 , aO 3 / with the terminal condition X.t of the backward model (12.10) is related to the drift term a D .a1 , a2 , a3 / of the forward model (12.6) via N t/ aO i .t , x, uE / D ai .t , x, uE /  C0 ".x,

@ ln pE .E u; x, t /. @ui

(12.11)

This form of the drift term is the consequence of Thomson’s well-mixed condition (see [237]). It ensures the relation between the forward and backward p. d. f.’s used in the construction of backward algorithms in Section 3.3. Note that in Appenix B at the end of the chapter such a relation is given for a more general case. Remark 12.1. In (12.10), the differential d W means that here the backward Ito integral is taken1. From this, the finite-difference form of the backward Ito equation (12.10) reads O 0 /  X.t O 0  t0 / D V.t O 0 /t0 , X.t O 0  t0 / O 0 /  V.t V.t O 0 /, V.t O 0 //t0 C D aO .t0 , X.t

q   O 0 /, t0 / W .t0 /  W .t0  t0 / , C0 ". N X.t

where the integration step t0 is positive. Thus in this chapter we will deal with the construction of Monte Carlo estimators for the integral (12.9) based on the simulation of forward and backward Lagrangian trajectories.

12.3

Monte Carlo estimators for the mean concentration and fluxes

In this section we construct Monte Carlo estimators for the mean concentration and fluxes at a fixed point and for integrals over space and time of these mean fields. In Section 12.3.1 we deal with forward estimators for the general case of nonstationary, possibly horizontally inhomogeneous turbulence. In Section 12.3.2 we modify these estimators to the horizontally homogeneous turbulence. Backward estimators are suggested in Section 12.3.3. 1

The backward Ito integral is defined by Z Z t . / d W . / :D s

T s T t

.T   / d WT . /,

s  t  T , WT . / :D W .T /  W .T   / is a standard Wiener process. This integral does not depend on the choice of T . For details see, e. g., [105].

244

Chapter 12 Evaluation of mean concentration and fluxes

12.3.1 Forward estimator Calculation of the mean concentration and fluxes at a fixed point by forward simulation is generally not possible (e. g., [57]). However if calculation of an integral of hg.E u.x, t //c.x, t /i over space and time is desired, Z Z T ˝ ˛ dx dt g.E u.x, t //c.x, t / H.x, t /, (12.12) IH D D

0

where T > 0 and H.x, t / is a weight function defined on D Œ0, T , then the forward estimator can be successfully used. As an example we mention the problem of evaluation of the center and size of a cloud. Let us give now a forward Monte Carlo estimator for the integral (12.12) with arbitrary function H.x, t /. Substituting (12.9) into the right-hand side of (12.12), we get Z T Z dx dt hg.E u.x, t //c.x, t /i H.x, t / D

0 T

Z

D

Z

Z dt0

0

Z

d uE

d x0 g.E u/ Q.x0 , t0 / D 1

Z

dt0

d x0 Q.x0 , t0 / D

0

dt

dx

dt t0

D

Z

T

IR3

f

Z

T

D

d uE 0 pE .E u 0 ; x 0 , t0 /

Z

d H.x, t /pL .x, uE , , t ; x0 , uE 0 , t0 /

0

Z

IR3

Z

IR3

Z d uE 0 pE .E u0 ; x0 , t0 /IEx0 ,uE 0 ,t0

T t0

xt 0 ,uE 0 ,t0 g.Vxt 0 ,uE 0 ,t0 /H.Xxt 0 ,uE 0 ,t0 , t /,

where IEx0 ,uE 0 ,t0 is the expectation over samples of stochastic processes Xxt 0 ,uE 0 ,t0 ,

Vxt 0 ,uE 0 ,t0 , xt 0 ,uE 0 ,t0 (for fixed x0 , uE 0 , t0 ). The forward estimator can be obtained by applying the randomization procedure (see e. g., [143, 191]) to the integrals (over t0 , x0 and uE 0 ) in the second line of the last equality. Randomization can be done by choosing an arbitrary p. d. f. r.x, t / defined in D Œ0, T , which is consistent with Q.x, t / in the sense that r.x, t / ¤ 0 if Q.x, t / ¤ 0. If .x0 , t0 / is a random point in D Œ0, T  with the p. d. f. r.x, t /, and uE 0 is a 3D random u; x0 , t0 /, then variable with the p. d. f. pE .E Z

Z

T

dt < g.E u.x, t //c.x, t / > H.x, t / D IEH ,

dx D

0

where H D

Q.x0 , t0 / r.x0 , t0 /

Z

T t0

dt xt 0 ,uE 0 ,t0 g.Vxt 0 ,uE 0 ,t0 /H.Xxt 0 ,uE 0 ,t0 , t /,

(12.13)

245

Section 12.3 Monte Carlo estimators for the mean concentration and fluxes

and IE stands for the expectation over ensemble of trajectories Xxt 0 ,uE 0 ,t0 , Vxt 0 ,uE 0 ,t0 ,

xt 0 ,uE 0 ,t0 with random initial points x0 , uE 0 , t0 . It is reasonable to choose r.x, t / proportional to Q.x, t /. In this case the factor Q=r in (12.13) is a constant, and this might result in a variance reduction.

12.3.2

Modified forward estimators in case of horizontally homogeneous turbulence

Time averaged mean characteristics In this subsection the turbulence is assumed to be horizontally homogeneous, generally nonstationary, and the coefficient of absorption does not depend on the horizontal coordinates: .x, t / D .z, t /. RT We use the horizontal homogeneity to calculate the time averaged mean 0 dt hg.E u.x, t //c.x, t /ih.t / at a fixed point x D .x, y, z/. Here h.t / is a weight function defined on Œ0, T . From the horizontal homogeneity it follows that f

f

pL .x, y, z, uE , , t ; x0 , y0 , z0 , uE 0 , t0 / D pL .x  x0 , y  y0 , z, uE , , t ; 0, 0, z0 , uE 0 , t0 /. u0 ; x0 , t0 / D pE .E u0 ; z0 , t0 /, we get Taking into account that pE .E Z

T

dt hg.E u.x, t //c.x, t /i h.t / 0

Z

Z

T

D 0

Z

0 T

Z D

dz0 0



IR3

d uE 0 pE .E u 0 ; z 0 , t0 /

1

Z

T

dt h.t / t0

d x0

D

Z IR3

d uE

d ı.z  z 0 /g.E u/ Q.x  x 0 , y  y 0 , z0 , t0 /pL .x0 , uE , , t ; 0, 0, z0 , uE 0 , t0 / f

Z dt0

0

Z

Z

1

dt0

(12.14)

Z

1

dz0 0

IR3

Z d uE 0 pE .E u0 ; z0 , t0 /IEz0 ,uE 0 ,t0

g.Vzt 0 ,uE 0 ,t0 / zt 0 ,uE 0 ,t0 Q.x



X tz0 ,uE 0 ,t0 , y



T t0

dt h.t /ı.z  Z tz0 ,uE 0 ,t0 /

Y tz0 ,uE 0 ,t0 , z0 , t0 /,

where IEz0 ,uE 0 ,t0 is the expectation over ensemble of trajectories Xzt 0 ,uE 0 ,t0 , Vzt 0 ,uE 0 ,t0 ,

zt 0 ,uE 0 ,t0 , t  t0 , which are defined as Xzt 0 ,uE 0 ,t0 D .X tz0 ,uE 0 ,t0 , Y tz0 ,uE 0 ,t0 , Z tz0 ,uE 0 ,t0 / D Xxt 0 ,uE 0 ,t0 , Vzt 0 ,uE 0 ,t0 D .U tz0 ,uE 0 ,t0 , V tz0 ,uE 0 ,t0 , W tz0 ,uE 0 ,t0 / D Vxt 0 ,uE 0 ,t0 , zt 0 ,uE 0 ,t0 D

.t ; x0 , t0 /, with x0 D .0, 0, z0 /. Now we will use the following property of the Dirac delta function (e. g., see [252, p. 36, eq. (9.3)]): for arbitrary continuous function f . / and continuously differen-

246

Chapter 12 Evaluation of mean concentration and fluxes

tiable function Z. / Z

t

f . /ı.Z. /  z/ d  D 0

X t .z/ j D1

f .j / ˇ ˇ, ˇ dZ. j / ˇ ˇ d ˇ

(12.15)

where  t .z/ is the number of intersections of the level z by the trajectory Z. / in the interval 0    t , and j are the intersection times. Thus, from (12.14), taking into account (12.15) we find Z T dt hg.E u.x, t //c.x, t /i h.t / (12.16) 0

Z

Z

T

D

dt0 0

Z

1

dz0 0

 t0 ,T .z/



X

h.j /

IR3

d uE 0 pE .E u0 ; z0 , t0 /IEz0 ,uE 0 ,t0

g.Vz j0 ,uE 0 ,t0 /

j D1

jW zj0 ,uE 0 ,t0 j

z j0 ,uE 0 ,t0 Q.x  X zj0 ,uE 0 ,t0 , y  Y zj0 ,uE 0 ,t0 , z0 , t0 /,

where  t0 ,T .z/ is the number of intersections of the level z by the trajectory Z tz0 ,uE 0 ,t0 in the interval t0  t  T , and j are the intersection times. Now the randomization of integrals over t0 , z0 and uE 0 in the right-hand side of the last equality enables us to obtain the final estimator. For this, consider a p. d. f. r.z0 , t0 / on Œ0, 1/ Œ0, T  which is consistent with the source Q.x, y, z, t / in the sense that r.z0 , t0 / ¤ 0 if there exist x, y such that Q.x, y, z0 , t0 / ¤ 0. Then, Z

T

dt hg.E u.x, t //c.x, t /ih.t / D IE1 .x/, 0

where 1 .x/ D

 t0 ,T .z/ X g.Vz j0 ,uE 0 ,t0 / z ,uE ,t 1 h.j /

j0 0 0 r.z0 , t0 / jW z0 ,uE 0 ,t0 j j D1

j

Q.x  X zj0 ,uE 0 ,t0 , y  Y zj0 ,uE 0 ,t0 , z0 , t0 /. Here z0 , t0 is a 2D random variable chosen from Œ0, 1/ Œ0, T  with the p. d. f. r.z0 , t0 /, u0 ; z0 , t0 /. and uE 0 is a 3D random variable with the p. d. f. pE .E Crosswind and time-averaged mean characteristics In this subsection the turbulence is assumed to be horizontally homogeneous (generally nonstationary), and the coefficient of absorption does not depend on the horizontal

247

Section 12.3 Monte Carlo estimators for the mean concentration and fluxes

coordinates : .x, t / D .z, t /. Let us estimate the crosswind and time-averaged mean characteristic Z

Z

T

Ih D

1

dt 1

0

dy hg.E u.x, y, z, t //c.x, y, z, t /i h.y, t /

at a fixed point .x, z/. Here h.y, t / is a weight function defined on .1, 1/ Œ0, T . Using the same arguments as in the previous subsection we get Z T Z 1 dt dy hg.E u.x, y, z, t //c.x, y, z, t /i h.y, t / (12.17) Ih D 1 1

0

Z

Z

T

D

dt0 0

Z



1

Z

dz0 0

T t0

Z

1

dy0 y0 ,z0 ,u E 0 ,t0

dt h.Y t

y ,z0 ,u E 0 ,t0

Q.x  X t 0

IR3

d uE 0 pE .E u0 ; z0 , t0 /IEy0 ,z0 ,uE 0 ,t0 y ,z0 ,u E 0 ,t0

, t /ı.z  Z t 0

y ,z0 ,u E 0 ,t0

/g.V t 0

y ,z0 ,u E 0 ,t0

/ t 0

, y0 , z0 , t0 /, y ,z0 ,u E 0 ,t0

where IEy0 ,z0 ,uE 0 ,t0 is the expectation over the ensemble of trajectories X t 0

y ,z ,u E ,t Vt 0 0 0 0 ,

y ,z ,u E ,t

t 0 0 0 0 ,

t  t0 , which are defined as

y ,z0 ,u E 0 ,t0

D .X t 0

y ,z0 ,u E 0 ,t0

D .U t

Xt 0 Vt 0

,

y ,z0 ,u E 0 ,t0 y0 ,z0 ,u E 0 ,t0

y0 ,z0 ,u E 0 ,t0

, Yt

y0 ,z0 ,u E 0 ,t0

, Vt

y ,z0 ,u E 0 ,t0

, Zt 0

/ D Xxt 0 ,uE 0 ,t0 ,

y0 ,z0 ,u E 0 ,t0

, Wt

y ,z0 ,u E 0 ,t0

/ D Vxt 0 ,uE 0 ,t0 , t 0

D .t ; x0 , t0 /, with x0 D .0, y0 , z0 /. From (12.17) we get by the property (12.15) that Z T Z 1 Z 1 Z dt0 dy0 dz0 d uE 0 pE .E u0 , z0 , t0 / IEy0 ,z0 ,uE 0 ,t0 Ih D 1

0

 t0 ,T .z/



X

j D1

0

h.Y yj0 ,z0 ,uE 0 ,t0 , j /

(12.18)

IR3

y ,z0 ,u E 0 ,t0

g.V j0

/

y ,z ,u E ,t jW j 0 0 0 0 j

y j0 ,z0 ,uE 0 ,t0 Q.x  X yj0 ,z0 ,uE 0 ,t0 , y0 , z0 , t0 /, y ,z ,u E ,t

where  t0 ,T .z/ is the number of intersections of the level z by the trajectory Z t 0 0 0 0 during the interval t0  t  T , and j are the intersection times. Now it is not difficult to construct a random estimator for Ih by applying a standard Monte Carlo randomization procedure for evaluation of integrals. In our case we apply it to the multiple integral in (12.18) over t0 , y0 , z0 and uE 0 . To this end, we consider a probability density r.y0 , z0 , t0 / on .1, 1/ Œ0, 1/ Œ0, T  which is consistent with

248

Chapter 12 Evaluation of mean concentration and fluxes

the source Q.x, y, z, t / in the sense that r.y0 , z0 , t0 / ¤ 0 if there exists x such that Q.x, y0 , z0 , t0 / ¤ 0. Then Z IE2 .x, z/ D

Z

T

1

dt 1

0

dy hg.E u.x, y, z, t //c.x, y, z, t /i h.y, t /,

where 2 .x, z/ D

 t0 ,T .z/ y ,z ,u E ,t X g.V j0 0 0 0 / y0 ,z0 ,uE 0 ,t0 1

j h.Y yj0 ,z0 ,uE 0 ,t0 , j / y ,z ,u E ,t r.y0 , z0 , t0 / jW 0 0 0 0 j j D1

j

Q.x  X yj0 ,z0 ,uE 0 ,t0 , y0 , z0 , t0 /. Here y0 , z0 , t0 is a random variable chosen in .1, 1/ Œ0, 1/ Œ0, T  with the u0 ; z0 , t0 /. density r.y0 , z0 , t0 /, and uE 0 is a 3D random variable with the p. d. f. pE .E Stationary turbulence In this subsection the turbulence is assumed to be horizontally homogeneous and stationary, and the coefficient of absorption depends only on height: .x, t / D .z/. In addition, the initial concentration is assumed to be zero: q0 .x/ D 0. These assumptions allow us to estimate hg.E u.x, t //c.x, t /i directly at a fixed point .x, t /. Indeed, under the assumed conditions f

f

pL .x, y, z, uE , , t ; x0 , y0 , z0 , uE 0 , t0 / D pL .x x0 , y y0 , z, uE , , t t0 ; 0, 0, z0 , uE 0 , 0/. u0 ; x0 , t0 / D pE .E u0 ; z0 / we get Therefore, by pE .E hg.E u.x, t //c.x, t /i Z t Z Z 1 Z Z dz0 d uE 0 pE .E u0 ; z0 / d d x0 D IR3

d uE

(12.19) 1

d ı.z  z 0 /g.E u/

0 D f 0 q.x  x , y  y , z0 , t   /pL .x , uE , ,  ; 0, 0, z0 , uE 0 , 0/ Z 1 Z Z t dz0 d uE 0 pE .E u0 ; z0 /IEz0 ,uE 0 d  ı.z  Z z0 ,uE 0 /g.Vz 0 ,uE 0 / z 0 ,uE 0 0 0 IR3 q.x  X z0 ,uE 0 , y  Y z0 ,uE 0 , z0 , t   /, 0

0

D

IR3

Z

0

0

where IEz0 ,uE 0 is the expectation over the ensemble of trajectories Xz 0 ,uE 0 , Vz 0 ,uE 0 , z 0 ,uE 0 ,

  0, which are defined as Xz 0 ,uE 0 D .X z0 ,uE 0 , Y z0 ,uE 0 , Z z0 ,uE 0 / D X x0 ,uE 0 ,0 , Vz 0 ,uE 0 D .U z0 ,uE 0 , V z0 ,uE 0 , W z0 ,uE 0 / D Vx 0 ,uE 0 ,0 , z 0 ,uE 0 D . ; x0 , 0/, with x0 D .0, 0, z0 /.

249

Section 12.3 Monte Carlo estimators for the mean concentration and fluxes

Now, from (12.19) we get by (12.15) Z 1 Z X t .z/ g.Vz j0 ,uE 0 / z ,uE hg.E u.x, t //c.x, t /i D dz0 d uE 0 pE .E u0 , z0 /IEz0 ,uE 0

j0 0 z0 ,u E0 0 IR3 jW j j j D1 q.x  X zj0 ,uE 0 , y  Y zj0 ,uE 0 , z0 , t  j /, where  t .z/ is the number of intersections of the level z by the trajectory Z z0 ,uE 0 during the interval 0    t , and j are the intersection times. We apply here the standard Monte Carlo randomization procedure to evaluate the integrals over z0 and uE 0 . To this end, we consider a probability density r.z0 / on Œ0, 1/ which is consistent with the source q.x, y, z, t / in the sense that r.z0 / ¤ 0 if there exist x, y, t such that q.x, y, z0 , t / ¤ 0. Then hg.E u.x, t //c.x, t /i D IE3 .x, t /, where 3 .x, t / D

 t .z/ g.Vz j0 ,uE 0 / z ,uE 1 X

j0 0 q.x  X zj0 ,uE 0 , y  Y zj0 ,uE 0 , z0 , t  j /. (12.20) z0 ,u E0 r.z0 / j j D1 jW j

Here z0 is a random variable chosen in Œ0, 1/ with the density r.z0 /, and uE 0 is a 3D u0 ; z0 /. random variable with the p. d. f. pE .E Analogously the crosswind averaged mean can be evaluated at a fixed point .x, z, t /: R1 dy hg.E u .x, y, z, t //c.x, y, z, t /i h.y/. Here h.y/ is a weight function defined on 1 .1, 1/. Let r.y0 , z0 / be a probability density defined on .1, 1/ Œ0, 1/ which is consistent with the source q.x, y, z, t / in the sense that r.y0 , z0 / ¤ 0 if there exist x, t such that q.x, y0 , z0 , t / ¤ 0. Then Z 1 ˝ ˛ dy g.E u.x, y, z, t //c.x, y, z, t / h.y/ D IE4 .x, z, t /, 1

where 4 .x, z, t / D

X y ,z ,u E t .z/ g.V j0 0 0 / y0 ,z0 ,uE 0 1

j h.Y yj0 ,z0 ,uE 0 / y ,z ,u E r.y0 , z0 / jW 0 0 0 j j D1

q.x 

X yj0 ,z0 ,uE 0 , y0 , z0 , t

j

 j /.

Here .y0 , z0 / is a random variable chosen in .1, 1/ Œ0, 1/ with the density y ,z0 ,u E0

r.y0 , z0 /, uE 0 is a 3D random variable with the p. d. f. pE .E u0 ; z0 / and X 0

,

y ,z ,u E y ,z ,u E y ,z ,u E y ,z ,u E W 0 0 0 , 0 0 0 (  0) are stochastic processes defined as X 0 0 0 D .X 0 0 0 , y ,z ,u E y ,z ,u E y ,z ,u E y ,z ,u E y ,z ,u E y ,z ,u E Y 0 0 0 , Z 0 0 0 / D Xx 0 ,uE 0 ,0 , V 0 0 0 D .U 0 0 0 , V 0 0 0 , W 0 0 0 / D y ,z ,u E V x0 ,uE 0 ,0 , 0 0 0 D . ; x0 , 0/, with x0 D .0, y0 , z0 /.

250

Chapter 12 Evaluation of mean concentration and fluxes

12.3.3 Backward estimator Unlike the forward algorithm, the backward technique enables us to estimate the mean concentration and fluxes at a fixed point in space and time, even in the general case of nonstationary turbulence. Therefore, the estimation can be done directly for hg.E u/ci. u/ci D hci or h g.E u/ci D hui ci, Note that taking g.E u/ equal to 1 or to ui , we get h g.E respectively. Analogously to forward Lagrangian p. d. f. (12.7), the backward Lagrangian p. d. f. can be defined as ® ¯ E E b u,t E O x,u,t O x,u,t .x0 , uE 0 , 0 , t0 ; x, uE , t / D IEx,u,t u0  V O x, pL E ı.x0  X t0 /ı.E t0 /ı. 0 

t0 / , u,t E is defined by where .t O 0 / D O x, t0

d .t O 0/ u,t E O x, D .X O 0 /, t0 , t0 / .t dt0

.t O / D 1,

E E O x,u,t O x,u,t and IEx,u,t E means the expectation over samples of stochastic processes X t0 , V t0 ,

u,t E E , 1. In appendix C it is shown and O x, t0 , t0  t , starting at final time t0 D t at point x, u that f

b pE .E u0 ; x0 , t0 /pL .x, uE , , t ; x0 , uE 0 , t0 / D pE .E u; x, t /pL .x0 , uE 0 , , t0 ; x, uE , t /. (12.21) Substituting the right-hand side of this equality to the right-hand side of (12.9), we get

hg.E u.x, t //c.x, t /i Z Z 1 Z t Z Z d uE d

dt0 d x0 g.E u/ Q.x0 , t0 / D IR3

0

0

D

IR3

d uE 0 pE .E u; x, t /

b .x0 , uE 0 , , t0 ; x, uE , t / pL Z Z t E u,t E O x,u,t D d uE pE .E u; x, t /g.E u/IEx,u,t dt0 O x, E t0 Q.X t0 , t0 /. IR3

0

From the last expression, using the standard Monte Carlo arguments, one gets O t /, hg.E u.x, t //c.x, t /i D IE.x, where O t / D g.E .x, u/

Z 0

t

(12.22)

E u,t E O x,u,t dt0 O x, t0 Q.X t0 , t0 /

 Z t x,u,t E x,u,t E u,t E x,u,t E O O q . X / C dt

O q. X , t / . D g.E u/ O x, 0 0 t0 0 0 t0 0 0

u; x, t /. Here uE is 3D random variable with the p. d. f. pE .E

(12.23)

Section 12.4 Application to the footprint problem

251

Now we are in a position to construct a Monte Carlo estimator for the integral (12.12) from hg.E u/ci over space and time with an averaging function H.x, t /. For this, we consider an arbitrary p. d. f. p.x, t / defined in D Œ0, T  which is consistent with H.x, t / in the sense that p.x, t / ¤ 0 if H.x, t / ¤ 0. Let .x, t / be a random point in D Œ0, T  with the p. d. f. p.x, t /, and uE be a 3D random variable with the p. d. f. u; x, t /. Then from (12.22)–(12.23) and the standard Monte Carlo arguments it pE .E follows that the random variable Z t H.x, t / E u,t E O x,u,t OH D dt0 O x, g.E u/ t0 Q.X t0 , t0 / p.x, t / 0  Z t H.x, t / x,u,t E x,u,t E x,u,t E x,u,t E O O D dt0 O t0 q.X t0 , t0 / g.E u/ O 0 q0 .X0 / C p.x, t / 0 is a Monte Carlo estimator for the integral (12.12): Z Z T dx dt hg.E u.x, t //c.x, t /iH.x, t / D IEOH . D

12.4

0

Application to the footprint problem

The footprint problem as formulated in the literature (e. g., see [55, 221]) essentially deals with the calculation of the contribution to the mean concentration and its flux at a fixed point from a surface source of a scalar. Let us consider a surface source at a height zs , and let F .x, y, t / be an amount of emitted scalar per unite time and area (at time t near the surface point .x, y/). Then the distribution function q.x, t / has the form q.x, t / D q.x, y, z, t / D F .x, y, t / ı.z  zs /.

(12.24)

We assume that the turbulence is horizontally homogeneous and stationary. The coefficient of absorption is assumed to depend only on height: .x, t / D .z/. The initial concentration distribution is assumed to be zero: q0 .x/ D 0. The Lagrangian trajectories are perfectly reflected at roughness height z . Therefore we will naturally assume that zs  z . First, let us construct a Monte Carlo estimator for hg.E u.x, t //c.x, t /i based on the zs ,u E0 zs ,u E0 zs ,u E0 forward Lagrangian trajectory X , V , ,   0 (see Section 3.2.3). Indeed, choosing in (12.20) r.z0 / D ı.z0  zs / and taking into account (12.24), we have hg.E u.x, t //c.x, t /i  X t .z/ g.Vz js ,uE 0 / z ,uE zs ,u E0 zs ,u E0 s 0

j F .x  X j , y  Y j , t  j / , (12.25) D IEzs ,uE 0 zs ,u E0 j j D1 jW j

252

Chapter 12 Evaluation of mean concentration and fluxes

where uE 0 is a 3D random variable with the p. d. f. pE .E u0 ; zs / and IEzs ,uE 0 means an

expectation over samples of the stochastic processes Xz s ,uE 0 , Vz s ,uE 0 , z s ,uE 0 ,   0. In practical implementation the mathematical expectation on the right-hand side of (12.25) is approximately calculated as

IEzs ,uE 0

 X t .z/ j D1

g.Vz js ,uE 0 /



z js ,uE 0 F .x  X zjs ,uE 0 , y  Y zjs ,uE 0 , t  j /

jW zjs ,uE 0 j N i g.Vi ij / i 1 XX '

F .x  X iij , y  Y iij , t  ij /, N jW iij j ij

(12.26)

iD1 j D1

where i denotes the trajectory starting with the initial velocity uE 0i (which is random u0 ; zs / and independent for different i ), N is the number of trawith the p. d. f. pE .E jectories, i is the number of intersections of the level z by i -th trajectory, and ij are the intersection times. Now let us construct a Monte Carlo estimator for hg.E u.x, t //c.x, t /i based on the x,u,t E x,u,t E u,t E O t , and O x, Ot , V backward Lagrangian trajectory X t0 , t0  t (see Section 3.3). 0 0 First we will assume that zs > z . Taking into account (12.24) and using the property (12.15) of the Dirac delta function, from (12.22)–(12.23) we find  Z t x,u,t E x,u,t E O dt

O q. X , t / g.E u / hg.E u.x, t //c.x, t /i D IEx,u,t 0 t0 0 E t0 0

 OX u,t E t .zs /

O x, j x,u,t E x,u,t E O O F .X j , Y j , j / , D IEx,u,t g.E u/ E E j jWO x,u,t j D1

(12.27)

j

where O t .zs / is the number of intersections of the level zs by the backward trajectory E in the interval 0    t ;  are the intersection times; u E is 3D random ZO x,u,t j u; x, t /, and IEx,u,t is the expectation taken over samples variable with the p. d. f. pE .E E

E E u,t E O x,u,t O x,u,t O x, of the stochastic processes X t0 , V t0 , and

t0 , t0  t . The surface emission at the height where the trajectories are reflected, (the case zs D z ) can be handled by letting E will zs ! z (zs > z ). Taking into account that for each time j the trajectory ZO x,u,t simultaneously pass twice (first in dawnward direction and, then, in upward one) the level zs , it is easy to establish that  Z t x,u,t E x,u,t E O dt0 O t0 q.X t0 , t0 / hg.E u.x, t //c.x, t /i D IE g.E u/ 0

 O X u,t E t .z /

O x, j x,u,t E x,u,t E O O F .X j , Y j , j / . D 2 IE g.E u/ E j jWO x,j u,t j D1

(12.28)

253

Section 12.5 Conclusion

In practice, the approximate calculation of mathematical expectations in the right-hand sides of (12.27)–(12.28) is carried out by similiar technique as in (12.26). For example, in the case zs D z we have  O X u,t E t .z /

O x, x,u,t E j O q.X j , j / IEx,u,t g.E u/ E E j jWO x,u,t j D1

j

'2

12.5

N O i

O i 1 XX E g.E u/ ij F .XO iij , XO x,iju,t , ij /. O i j N j W iD1 j D1 ij

(12.29)

Conclusion

Direct and backward Lagrangian stochastic algorithms for the numerical evaluation of the mean concentration of scalars and their fluxes were suggested and justified. The random estimators were constructed in the form of expectations over stochastic Lagrangian trajectories governed by Langevin-type equations derived from Thomson’s well-mixed condition. The transported scalar may be absorbed. Detailed expressions for random estimators for the mean characteristics (concentration, fluxes, time and space averages of concentration and fluxes) for quite general cases of sources were given. A practically important case of a plane source (related to the so-called “footprint problem”) was treated in detail. Advantages of the methods developed were that they are flexible to the structure of the source and the measured statistical characteristics.

12.6

Appendices

12.6.1

Appendix A. Representation of concentration in Lagrangian description

Here we show that the equality (12.1) is true. The total instantaneous concentration c.x, t / can be represented as the sum of c0 .x, t / and c1 .x, t /, defined by @c0 @c0 .x, t / C .x, t /c0 .x, t / D 0, t > 0; C ui .x, t / @t @xi

c0 .x, 0/ D q0 .x/,

.A1/

and @c1 @c1 .x, t / C .x, t /c1 .x, t / D q.x, t /, t > 0; C ui .x, t / @t @xi

c1 .x, 0/ D 0,

respectively. First we show that Z c0 .x, t / D d x0 .t ; x0 , 0/q0 .x0 /ı.x  X.t ; x0 , 0//. D

.A2/

254

Chapter 12 Evaluation of mean concentration and fluxes

According to (A1) the function C0 .t / D C0 .t ; x0 / D c0 .X.t ; x0 , 0/, t / satisfies the equation dC0 .t / C .X.t ; x0 , 0/, t /C0 .t / D 0, C0 .0/ D q0 .x0 /. dt Therefore from the definition of .t ; x0 , t0 / given by (2) it follows that C0 .t ; x0 / D

.t ; x0 , 0/q0 .x0 / and Z d x0 .t ; x0 , 0/q0 .x0 /ı.x  X.t ; x0 , 0// D Z D d x0 c0 .X.t ; x0 , 0/, t /ı.x  X.t ; x0 , 0// ZD D d y0 c0 .y0 , t /ı.x  y0 / D c0 .x, t /. D

Here in the last integral the substitution of variables x0 ! y0 D X.t ; x0 , 0/ was performed, and it was taken into account that the Jacobian of this transformation equals unity due to incompressibility of the velocity field uE .x, t / ( [145]). With this, (A2) is established. Now the following equality will be shown: Z t Z dt0 d x0 .t ; x0 , t0 /q.x0 , t0 /ı.x  X.t ; x0 , t0 //. .A3/ c1 .x, t / D D

0

Indeed, it can be easily shown (by taking suitable derivatives) that Z t c1 .x, t / D dt0 g t0 .x, t /,

.A4/

0

where g t0 .x, t /, .t0 > 0/ is defined by @g t @g t0 .x, t / C ui .x, t / 0 C .x, t /g t0 .x, t / D 0, t > t0 ; @t @xi

g t0 .x, t0 / D q.x, t0 /.

.A5/ From (A5) and by the definition of the function .t ; x0 , t0 / given by (12.2), it follows that g t0 .X.t ; x0 , t0 /, t / D .t ; x0 , t0 /q.x0 , t0 /. From this equality and since the Jacobian of the transformation x0 ! y0 D X.t ; x0 , t0 / is equal to unity, we obtain Z d x0 .t ; x0 , t0 /q.x0 , t0 /ı.x  X.t ; x0 , t0 // D Z D d x0 g t0 .X.t ; x0 , t0 /, t /ı.x  X.t ; x0 , t0 // D Z D d y0 g t0 .y0 , t /ı.x  y0 / D g t0 .x, t /. D

255

Section 12.6 Appendices

The last equality and (A4) yields (A3). Since c.x, t / D c0 .x, t / C c1 .x, t /, from (A2) and (A3) it follows that the representation (12.1) holds.

12.6.2

Appendix B. Relation between forward and backward transition density functions

Here we present the relation between the forward and backward p. d. f.’s used further y ,t in Appendix C. Let p f .y, t ; y0 , t0 / D hı.y  Y t 0 0 /i be the transition density function y ,t of the n-dimensional diffusion process Y t 0 0 , the solution to d Yi .t / D Ai .Y.t /, t /dt C ij .Y.t /, t /d Wj .t /, t > t0 , i D 1, : : : , n, Y.t /j tDt0 D y0 , .B1/ where Ai .y, t / and ij .y, t / are functions defined in D Œ0, T , W1 .t /, : : : , Wn .t / are independent standard Wiener processes, and D is a domain in IRn , T > 0. We assume that the boundary of D is impenetrable, i. e., the trajectories determined by (B1) do not reach the boundary. Assume that we have a positive function .y, t / defined on D Œ0, T  as a solution to the equation

1 @2 .Bij / @ @ .Ai / D , C @t @yi 2 @yi @yj

.B2/

y,t

where ik j k D Bij . Let p b .y0 , t0 ; y, t / D hı.y0  Z t0 /i be the transition density of y,t

the diffusion process Z t0 , 0  t0  t which is defined by dZi D Ai .Z, t0 / dt0 C ij .Z, t0 /, t0 / d Wj .t0 /,

t0 < t ,

Z.t / D y.

.B3/

Here d Wj .t0 / is defined as in the footnote to (12.10) in Section 12.2, and Ai .y, t / D Ai .y, t / 

1 @ .Bij .y, t /.y, t //. .y, t / @yj

We assume again, that the solutions to (B3) do never reach the boundary of D. Then the following relation is true (see [111], Appendix C): .y0 , t0 /p f .y, t ; y0 , t0 / D .y, t /p b .y0 , t0 ; y, t /.

12.6.3

.B4/

Appendix C. Derivation of the relation between the forward and backward densities

Here the derivation of the relation between forward and backward Lagrangian transition p. d. f.’s, equation (12.21), is presented. It is assumed that the boundary z D 0 is impenetrable, i. e., the trajectory, the solution to (12.6), will never reach this boundary.

256

Chapter 12 Evaluation of mean concentration and fluxes

Let us define C.t / D C tx0 ,uE 0 ,c0 ,t0 as the solution to dC.t / C .Xxt 0 ,uE 0 ,t0 , t /C.t / D 0, dt

t > t0 ;

C.t0 / D c0 ,

and the extended forward Lagrangian transition p. d. f. f

PL .x, uE , c, t ; x0 , uE 0 , c0 , t0 /

  u  Vxt 0 ,uE 0 ,t0 /ı.c  C tx0 ,uE 0 ,c0 ,t0 / , D IEx0 ,uE 0 ,c0 ,t0 ı.x  Xxt 0 ,uE 0 ,t0 /ı.E

where the forward trajectory Xxt 0 ,uE 0 ,t0 , Vxt 0 ,uE 0 ,t0 is defined in Section 12.2 and IEx0 ,uE 0 ,c0 ,t0

means an expectation over samples of stochastic processes Xxt 0 ,uE 0 ,t0 , Vxt 0 ,uE 0 ,t0 , and

C tx0 ,uE 0 ,c0 ,t0 starting at time t D t0 from the point x0 , uE 0 , c0 . Analogously, we define E as the solution to CO .t0 / D CO tx,0 u,c,t d CO .t0 / u,t E O x, O C .X t0 , t0 /C .t0 / D 0, dt0

t0 < t ;

CO .t / D c,

and the extended backward Lagrangian transition p. d. f. ® ¯ E E E O x,u,t O x,u,t O x,u,c,t ı.x  X uV / , PLb .x0 , uE 0 , c0 , t0 ; x, uE , c, t / D IEx,u,c,t E t0 /ı.E t0 /ı.c  C t0 E O x,u,t where IEx,u,c,t means the expectation over samples of stochastic processes X E t0 ,

E E O x,u,t O x,u,c,t V , t0  t , starting at final time t0 D t at point x, uE , c. t0 , C t 0 To derive the relation (12.21), first we establish the following equality:

pE .E u0 ; x0 , t0 / f u; x, t / b pE .E PL .x, uE , c, t ; x0 , uE 0 , c0 , t0 / D PL .x0 , uE 0 , c0 , t0 ; x, uE , c, t /. c0 c .C1/ To this end, we use the well-mixed condition [237]: @.ai pE / 1 @ 2 pE @.ui pE / @pE C D C0 "N.x, t / . C @t @xi @ui 2 @ui @ui Denote .x, uE , c, t / D

pE .E u; x, t / . c

From the well-mixed condition we get @2  @ @.ui / @.ai / @. .x, t / c / 1 C C . C D C0 "N.x, t / @t @xi @ui @c 2 @ui @ui Now (C1) follows from (C2) and from the result obtained in Appendix B.

.C2/

257

Section 12.6 Appendices

Using

f

f

pL .x, uE , c, t ; x0 , uE 0 , t0 / D PL .x, uE , c, t ; x0 , uE 0 , 1, t0 / and assuming in (C1) that c0 D 1 and c D , we get f

u0 ; x0 , t0 /pL .x, uE , , t ; x0 , uE 0 , t0 / D pE .E

pE .E u; x, t / b PL .x0 , uE 0 , 1, t0 ; x, uE , , t /.

.C3/

Further taking into account that x,u,,t E D CO t0

u,t E

O x, t0

,

and using the following property of the Dirac delta function ı.a  b/ D

b 1 ı.  /, a a

, a, b 2 .1, 1/,

u,t E with b D 1 and a D 1= O x, t0 , we get x,u,,t E ı.1  CO t0 /Dı



u,t E

O x, t0

u,t E u,t E u,t E  1 D O x, O x, O x, t0 ı. 

t0 / D ı. 

t0 /.

From this and the definition of the function PLb it follows that PLb .x0 , uE 0 , 1, t0 ; x, uE , , t / ® ¯ E E E O x,u,t O x,u,t D IEx,u,t /ı.E u0  V /ı. 0  O x,u,t / E ı.x0  X t0

D

t0

t0

b .x0 , uE 0 , 0 , t0 ; x, uE , t /.

pL

Substitution of the right-hand side of the last equality into (C3) completes the proof of the relation (12.21).

Chapter 13

Stochastic Lagrangian footprint calculations over a surface with an abrupt change of roughness height

Forward and backward stochastic Lagrangian trajectory simulation methods are developed to calculate the footprint and cumulative footprint functions of concentration and fluxes in the case when the ground surface has an abrupt change of the roughness height. The statistical characteristics to the stochastic model are extracted numerically from a closure model which we developed for the atmospheric boundary layer. The flux footprint function is perturbed in comparison with the footprint function for surface without change in properties. The perturbation depends on the observation level as well as roughness change and distance from the observation point. It is concluded that the footprint function for horizontally homogeneous surface, widely used in estimating sufficient fetch for measurements, can be seriously biased in many cases of practical importance.

13.1 Introduction Over a horizontally homogeneous surface the flux measured by the micrometeorological technique equals the surface flux. This principle is used to determine the surface exchange by the eddy covariance (EC) technique. The flux footprint function (e. g., see [221]) links the surface emissions to the observed fluxes above surface at EC measurement level. The footprint function is therefore used to estimate a distance required to make reliable EC measurements, i. e., if the horizontal extent of the underlying surface of interest is sufficient to determine its exchange rate. Extended tower measurements of fluxes over forests have been used during the past ten years to obtain detailed information on carbon and water exchanges between forest canopies and the atmosphere [90, 187, 246]). Large areas of forest are, however, common neither in Europe nor in the US. The footprint models based on analytical diffusion theory [79,221,225] as well as the Lagrangian stochastic simulation of the ensemble of fluid parcel trajectories [6,55,126] assume a horizontally homogeneous surface. For forest canopies the footprint models involve a number of uncertainties originating from the parametrization of the canopy turbulence features [174]. Such models are frequently applied in order to estimate the contribution of an area at a certain upwind distance, or to estimate the fetch to ensure that the given area contributes a certain percent to the observed

Section 13.2 The governing equations

259

flux, by vaguely assuming that the footprint function for a horizontally homogeneous surface is a good approximation for more complex situations with changes in surface properties. In reality, changes in surface roughness can be very drastic, for example in the case of forest and field interface. Also the thermal inhomogeneities induced by albedo and the repartitioning of available energy into sensible and latent heat fluxes can be significant; this will be analyzed, however, in the second part of this chapter.Here we will deal with pure mechanical turbulence caused by the surface roughness. One additional remark here should be made. Over rough surfaces, such as tall vegetation, vertical displacement of surface layer profiles occurs relative to the ground surface. Height displacement is usually 2/3 to 3/4 of the height of the roughness elements, whereas the roughness length is usually 1/30 to 1/10 of the vegetation height. In this study the flow inside vegetation is not considered, and the observation level (detector height) is equivalent to that of relative to the displacement height in a real measurement setup. Displacement height would effectively elevate flow streamlines and not affect the results qualitatively, whereas roughness change induces transition in horizontal wind speed and also, via mass conservation, nonzero vertical winds. Lagrangian trajectory simulation can be used in the inhomogeneous flow field (e. g., see [237], and [115]). However, to make the stochastic trajectory simulation possible, the mean flow and some other statistical moments have to be found. We will extract this data from a closure model, conventionally obtained from the Reynolds-averaging equations. The footprint function for inhomogeneous surface is estimated by backward Lagrangian trajectory simulation, and the perturbations relative to footprint function for horizontally homogeneous case are analyzed.

13.2

The governing equations

There exists a variety of closure models for turbulent mixing, ranging from constant eddy coefficient parametrization to detailed large eddy simulations and direct numerical simulation. As mentioned in [5], the performance of a k-model is almost identical to that of the k–"-model. We assume that the mean profiles in the boundary layer of atmosphere are described by the following system [253]: u

@u @ @u @u Cw D k C f .v  G sin ˛/ @x @z @z @z (13.1)

@v @v @ @v u Cw D k  f .u  G cos ˛/ @x @z @z @z is the momentum equation, where the Cariolis parameter is given by f D 2 sin ',  D 7.29 105 s1 , G is the geostrophical wind, and ' is the angle between the x-axis and the isobare; ˛ is the angle between the x-axis and the direction of the geostrophical

260

Chapter 13 Lagrangian footprint calculations

wind. Further, @u @w C D0 @x @z is the continuity equation, where k is the turbulent exchange coefficient for the momentum. The balance of the kinetic energy is written as @b @ @b @b Cw D ˛b k Ck u @x @z @z @z



@u @z



2 C

@v @z

2   "N,

(13.2)

2

where "N D cbk is the mean rate of dissipation of the turbulent energy, c D 0.0286, ˛b  0.7. The functions k and b are related through  lD

1 1 C z l0

1 ,

p k D Ck l b,

(13.3)

with  0.4, l0 D 100 m, Ck D 0.41. We will deal in this first part of the chapter with purely mechanical turbulence, and our system of governing equations consists of (13.1), (13.2), and the relation (13.3). The functions vary in the layer z0  z  h, h being the height of the boundary layer, and z0 the roughness height. The system of equations is considered with the following boundary conditions: u D 0,

v D 0,

u D G cos ˛,

at z D z0 ,

v D G sin ˛

@v @u D D 0 at @z @z @b D0 @z

w D 0,

at z D h, x  0,

z D h, x > 0,

at z D z0 ,

and

b D 0 at

z D h,

At z D z0 we take l D z0 .

13.2.1 Evaluation of footprint functions We assume that our 3D flow is homogeneous in the y-direction and inhomogeneous in the z- and x-directions. Roughness inhomogeneity along the x-direction is only assumed in our numerical analysis. The flow is considered in the boundary layer of height h, with a roughness height z0 D z01 which is a constant for x < 0, then in a small interval 0 < x <  it

261

Section 13.2 The governing equations

linearly changes from z01 to z02 , and then it is again constant: for x >  it equals z02 . The detector is placed at a point .xd , yd , zd /. We will deal with the footprint function of concentration c.x/ D c.x; xd , yd , zd / defined as the mean concentration at the detector point from a linear source with the coordinate x, placed at roughness height, directed along the axis Y. Analogously the footprint function of the vertical flux is defined as q.x/ D q.x; xd , yd , zd / D hw.xd , yd , zd / c 0 .x; xd , yd , zd /i, where c 0 is the fluctuating part of the concentration. The cumulative footprint functions of concentration and of vertical flux are defined as Z 1 Z 1 c.x 0 /dx 0 , Q.x/ D q.x 0 /dx 0 . C.x/ D x

x

The normalized footprint function of flux is defined as Qn .x/ D Q.x/=Q.1/. In this section we present the cumulative footprint functions for smooth-to-rough and rough-to-smooth changes of the roughness height. We calculate the mean concentration ci and qix , the vertical concentration fluxes at the detector point, from a surface source uniformly distributed over strips of width i D bi  ai : Di D ¹ x, z : ai  x  bi z D z0 , º, i D 1, : : : , nsr . The corresponding cumulative and normalized cumulative footprint functions of flux are also calculated. Calculations are carried out for small values of i , and the result is then normalized by the strip width. The forward simulation technique cannot be applied for estimation of concentration and flux exactly at a point in space in case of horizontally inhomogeneous turbulence, and/or exactly at moment t in case of non-stationary turbulence. Instead, one might consider averages over space and/or time localized near x and/or t (e. g., see [111]). However, such a simulation might be computationally inefficient if the extension of the source is much larger than that of the detector. For these cases, the backward trajectory simulation, starting from the space-time point of interest, is more efficient (in the case of Eulerian approach see [191], and in the Lagrangian framework see, e. g., [56]). To be more specific, let us present now the backward estimators for the evaluation of footprint functions in the case of the boundary layer with the sources uniformly distributed over the strips Di , i D 1, : : : , nsr . For simplicity, we have taken the xaxis coincident with the direction of the geostrophical wind, i. e., ˛ D 0. The backward trajectory starts at time t , at the detector point with the velocity sampled from the Eulerian velocity p. d. f. pE .u, x/ which is assumed to be Gaussian, see Appendix B at the end of this chapter. We note only that below, we denote by uN E k the k-th component (k D 1, 2, 3) of the mean Eulerian velocity vector, and the hat over the symbols x and u is to indicate that this is a finite-difference approximation to the true Lagrangian trajectory.

262

Chapter 13 Lagrangian footprint calculations

The backward trajectory simulation is conveniently carried out through the semiimplicit Euler scheme, which can be written for one time step as follows: O // t , xO k .t  t / D xO k .t /  .uO 0k .t / C uN E k .t , x.t uO 0k .t  t / D uO 0k .t /  aO k0 .t , x.t O  t /, u.t O // t p C C0 ".t N  t , x.t O  t // t  tk , where  tk , k D 1, 2, 3 are independent standard Gaussian random variables. Here for a reason of practical convenience, we work in the “primed” velocity variables uO 0k D uO k  uN E k , so that d xO k D .uO 0k C uN E k / ds, p d uO 0k D aO k0 ds C C0 "N d Wk .s/,

s < t , k D 1, 2, 3,

with the condition that the trajectory starts at the detector position with the velocity sampled from the Gaussian p. d. f. pE . Here according the formula given in Appendix B of this chapter we have @uN E k @uN E k 0 uN Ej  uO @xj @xj j C0 "N 1 @kj km km @im 0 0 @j m 0 D C uO C uO uO kj uOj0 C uN E i 2 2 @xj 2 @xi j 2 @xj j i D ˛O k C ˇOkj uO 0 C kj i uO 0 uO 0

aO k0 D aO k 

j

j

i

and ˛O k D

1 @kj , 2 @xj

C0 "N km @j m ˇOkj D , kj C uN E i 2 2 @xi

kj i D

km @im . 2 @xj

All the expressions for the input functions are given in the Appendices of this chapter. Recall that we use here the summation convention taking the sum over repeated indices i , j , l D 1, 2, 3, and d Wk .s/ stands for the backward Wiener differential (see [115]) which implies for the Euler scheme that the increments are taken back in time. O s  t reaches Let us denote by ij the time at which the trajectory .Ox.s/, u.s//, the ground surface and touches the i -th strip: the first touchdown at i1 , the second (after a reflection from the boundary) at i2 . etc., and the last one at iNi . The random estimators have the following form (see Appendix B), for the concentration: ci D

X Ni j D1

2 1 , i juO 3 .ij /j

263

Section 13.3 Results

and for the vertical flux: qiz D

X Ni j D1

2 uO 3 .t / , i juO 3 .ij /j

i D 1, : : : , nsr .

Here the angle brackets stands for the averaging over the ensemble of independent backward trajectories.

13.3

Results

In this section we will examine the impact of the roughness change on the footprint functions. The developed code calculates the footprint and cumulative footprint functions of concentration and flux for the horizontally inhomogeneous case when the roughness height is constant z01 for x < 0, then in a small interval 0 < x <  it linearly changes from z01 to z02 , and then it is again constant: for x >  it equals z02 . In calculations, we have taken  D jz02  z01 j. Note that when we speak of the inhomogeneous case, two essentially different cases are considered: z01 < z02 (smooth-to-rough) and z01 > z02 (rough-to-smooth). To be specific, in all calculations we have taken the geostrophical wind as G D 10 m/s, and the boundary layer height as h D 1 km. So the spatial scale in the figures is given in km. To give a sensitivity analysis to the change of these parameters, we present simultaneously the footprint functions of the inhomogeneous and homogeneous cases with the corresponding roughness height. These enables us to find the regions of applicability of the results obtained for the homogeneous case and, moreover, to conclude where the inhomogeneous case shows considerable differences compared to the homogeneous roughness. We also present some other footprints, in particular, the cumulative footprint functions of concentration and vertical flux. In all calculations we run 4 105 backward trajectories, the strip width was 2 m, the time step was varyN is the Lagrangian time scale ing according to t D 0.025 , where  D 2w2 =.C0 "/ at the trajectory instantaneous position. Since the variance of the random estimators was large, we have made a Gaussian smoothing procedure with a band width equal to 4 strip widths.

13.3.1

Footprint functions of concentration and flux

Let us describe the results of numerical simulations. We have made the calculations for two different cases: (i) smooth-to-rough and (ii) rough-to-smooth change of the roughness height. The detector was placed at the height zd D 2 m, at a distance xd.

264

Chapter 13 Lagrangian footprint calculations

Smooth-to-rough case In these calculations, along with the inhomogeneous case, we plot simultaneously the footprint functions for the homogeneous case with roughness height z0 D z02 . In Figure 13.1, left, we present the footprint function of concentration for the case of roughness change indicated above, for xd D 20 m, xd D 50 m and xd D 100 m (the same curve is given for the homogeneous case at z0 D 5 cm) as functions of the dimensionless upwind distance X= h, where the upwind distance X is defined as X D x C xd , the distance to the detector. The same curves are shown in the Figure 13.1, right, for the case z01 D 1 cm, z02 D 25 cm. It is seen that the footprint functions of concentration for inhomogeneous cases are all smaller compared to the homogeneous case in the near-region which is X= h < 0.04 in the left panel, and X= h < 0.08 in the right panel. In the far-region the situation is reverse: all the inhomogeneous curves are over the homogeneous curve after X= h > 0.15. The inhomogeneous curves have local minima at the position of roughness change; after this point they increase and become higher than the homogeneous curve. It is clearly seen that the minima are more pronounced in the case of larger roughness change. Note that this behavior explains why the homogeneous curve in the left panel of Figure 13.2 (the corresponding cumulative footprint function of concentration) is higher than all the curves for the small upwind distances, and then decreases down all the curves; here we give more detailed dependence on xd starting from xd D 20 m, with the last value xd D 200 m, where the influence of the roughness change is expected have almost disappeared. The same is true for the right panel (note that we have shown the curves only for distances X= h < 1, and therefore we cannot see the position where the homogeneous curve is below all the inhomogeneous curves, but in calculations this took place). In Figure 13.3 we present the corresponding footprint functions of flux, for the case of small (left panel) and larger roughness change (right panel). Note that here the difference between the homogeneous and inhomogeneous curves in the near-region is in the case of larger roughness change (right panel) much higher than that of smaller roughness change (left panel). In all cases the homogeneous curve is positioned almost everywhere below the inhomogeneous curves; one exception is in the neighborhood of the roughness change which is clearly seen in the right picture for the case xd D 20 m. In Figure 13.4 the cumulative footprint function of flux Q is presented. It is clearly seen that the difference between the homogeneous and inhomogeneous curves becomes less and less as the value of xd increases. Note that the homogeneous curve tends to 1 as the upwind distance increases, while in the inhomogeneous case the curves tend to asymptotic values which are larger than 1. To find the footprint area, it is convenient to use the normalized cumulative footprint function of flux, which is defined as the cumulative footprint function of flux divided by the corresponding asymptotic value. These curves are shown in Figure 13.5. To illustrate this, here we

Section 13.3 Results

265

show through a horizontal dashed line the level of 90 % contribution to the detector made by the surface around the detector position. Calculations show that the fetch in the homogeneous case is smaller than that of the inhomogeneous case; in Figure 13.5 it is seen that the corresponding fetch in the homogeneous case is about 200 m, while for xd D 20 m it is about two times larger. From this figure we can conclude that the inhomogeneous case with xd D 200 m is approximately coincident with the homogeneous case, and hence the analytical formulas known for the homogeneous case can be applied inside a region whose diameter is not less than 200 m. For xd < 200 m, the change of roughness should be taken into account. This confirms the known practical recommendation which says that the inhomogeneity can be neglected if xd =zd > 100. Rough-to-smooth case Here we present the calculations for the rough-to-smooth case showing the same footprint functions plotted in Figures 13.1–13.5. These footprint functions, when compared to the corresponding footprint functions presented in Figures 13.1–13.5, have the following features: the local minima at the roughness change in Figures 1,3 correspond to the local maxima in Figures 13.6 and 13.8. Also, in Figures 13.6 and 13.8 the homogeneous curve is below the inhomogeneous curves in the near-region, while in the far-region it is above these curves. The same is true for the cumulative footprint functions plotted in Figures 13.7 and 13.9. Note that in the inhomogeneous case, the asymptotic value of cumulative footprint functions of flux at large distances is less than 1; it is seen that the smaller xd is, the smaller is this asymptotic value. For larger roughness change it is becoming even less. As to the fetch, we can conclude from Figure 13.10 that, in contrast to the smoothto-rough case, here the fetch of homogeneous case is larger than that of the inhomogeneous cases; for instance, in the case of the roughness change z01 D 25 cm, z02 D 1 cm, the fetch is about 80 m for xd D 50 m, while for the homogeneous case it is about 300 m. Some features of the qualitative behavior of the footprint functions Let us describe some features of the qualitative behavior of the footprint functions for the inhomogeneous case. In Figure 13.11 we plot the homogeneous curves for two cases of the roughness height: z0 D 1 cm and z0 D 25 cm, and the inhomogeneous curves for the case of roughness change from z01 D 1 cm to z02 D 25 cm (left panel: the footprint of concentration; right panel: the footprint function of the flux). The position of the roughness change is shown by the dashed vertical line. First consider the results plotted in the left panel. In the near-region (left of the dashed vertical line) the inhomogeneous curve behaves qualitatively like the homogeneous curve for z0 D 25 cm, lying however considerably below with its maximum position shifted to the left (closer to the detector) when compared with the maximum

266

Chapter 13 Lagrangian footprint calculations

position of the homogeneous curve. In the region X= h > 0.08 we observe a qualitatively similar behavior of the inhomogeneous curve and the homogeneous curve, but for z0 D 1 cm; the curves are converging in the far-region. The maximum position of the inhomogeneous curve is shifted to the right. Thus the qualitative behavior of the inhomogeneous curve is controlled by the two homogeneous curves – in the closeregion by the case z0 D 25 cm, and in the far-region by the case z0 D 1 cm. This leads to the bimodal shape of the inhomogeneous curve. But from a simple superposition of the two homogeneous footprint functions, we would not expect such a deep drop between the two modes. This drop is caused by the flow structure around the change of roughness: in contrast to the homogeneous case here we have the positive vertical component of the mean velocity. Generally, the same arguments are true for the footprint function of flux shown in the right panel of Figure 13.11, with a not so deep drop between the two modes. This can be explained by the fact that the maximum position of the homogeneous curve with z0 D 1 cm is closer to the roughness jump. Note that from this picture we can clearly see that the cumulative footprint function of flux Q for the inhomogeneous case is larger than 1, in contrast to the homogeneous case, where it is always less than 1. Indeed, note that the area below the homogeneous curve z0 D 25 cm equals 1, so if we take the area below the homogeneous curve z0 D 25 cm in the region X < xd and add the area below the homogeneous curve z0 D 1 cm in the region X > xd , we get a value which is larger than 1, which follows from a simple comparison of the behavior of the curve. Let us turn to the rough-to-smooth case. In Figure 13.12 we plot the same curves as in Figure 13.11, but for the roughness change from z01 D 25 cm to z01 D 1 cm. At the roughness change position we observe a small jump in the inhomogeneous curve presenting the footprint function of concentration (left panel). Again, the general form of this curve can be deduced from the superposition of the two homogeneous curves, while the jump can be explained here by the negative values of the vertical component of the mean flow around the roughness change. An analysis of the footprint function of flux (right panel) analogous to that made for the smooth-to-rough case above shows that the cumulative footprint function of flux can here be less than 1. A larger change of the roughness height Further, we have made calculations for the larger change in roughness height, namely, for z01 D 1 cm, z02 D 100 cm in the smooth-to-rough case, and z01 D 100 cm, z02 D 1 cm in the rough-to-smooth case. The detector height was taken at zd D 20 m. The relevant footprint functions of concentration are shown in Figure 13.13, the footprint functions of flux q in Figure 13.14, and the cumulative footprint functions of flux Q in Figure 13.15, for different values of xd . The results are in good agreement with the conclusions made for the smaller change of the roughness height, showing even more

Section 13.3 Results

267

clearly the qualitative behavior of the curves discussed above. It should be noted that in this case we cannot expect a good quantitative prediction, because for the high roughness height we need to make a correction of the mean flow model which takes into account that the inhomogeneity affects the mean flow in a more complicated manner. Dependence on the detector height zd Let us now consider the dependence of the footprint functions on the detector height zd . In Figure 13.16 we present the footprint functions of concentration (left panel, in log-log scale) and flux (right panel, in log-line scale) for the smooth-to-rough case (z01 D 1 cm, z01 D 25 cm; the x-coordinate of the detector is fixed at xd D 100 m, and its height is varying from zd D 2 m to zd D 16 m. In the left panel it is seen that the local minima can be observed in all curves, but the higher the detector is, the smaller is the drop whose width is becoming larger with the height. The influence of the roughness change on the footprint function of flux is observable (right panel) until the height of about zd D 16 m. The corresponding cumulative footprint functions of concentration C (left panel) and flux Q (right panel) are shown in Figure 13.17. Note that the function C is uniformly decreased (in the considered region) with the height zd . For the function Q this is not the case: the curve for zd D 4 m first rapidly increases, being larger than all the other curves, and then slows down and finally tends to its asymptotic value at distances (about X= h  1) where the other curves are still increasing. Note that there is no monotonic behavior of the asymptotics Q.X= h/ at large values of X= h: first it increases with height zd , for zd D 4 m it is smaller than for zd D 8, while for zd D 32 m the value is the smallest compared to all the other values. This can be explained as follows. When the height zd is small, the detector is well inside the inner boundary layer generated by the roughness change, and the situation is close to the homogeneous case (with the roughness height z0 D z02 ) where the asymptotic value is 1. For large heights zd (larger than the height of the inner boundary layer) we are again in a situation close to the homogeneous case with the roughness height z0 D z01 , and therefore the corresponding asymptotics is again close to 1. In between, for intermediate heights, the asymptotic value is larger than 1 (e. g., for zd D 4, 8, 16 m). The additional contribution is coming from the trajectories with positive vertical velocities in the neighborhood of the position of the roughness change (the source is in a sense effectively lifted). The same arguments are true for the rough-to-smooth case, with the feature that in the intermediate heights the asymptotic values are less than 1, because in this case the trajectories get negative vertical velocities at the position of the roughness change. It should be noted that if we define the cumulative footprint function differently by omitting the convective part hwihci, i. e., as Q t D hw 0 c 0 i, then the above-mentioned asymptotics is almost always true for Q t . This holds for the smooth-to-rough case while for the rough-to-smooth case it is true only for large values of xd (xd > 20zd ); see Figure 13.18.

268

Chapter 13 Lagrangian footprint calculations

c

c homogeneous z = 5 cm

800

homogeneous, z0=25 cm

1200

0

700

1000

600 800

xd = 20 m

500

xd = 50 m

400

600

x =20 m d

x =100 m

300

d

400

x = 100 m d

200

200

x =50 m d

100 −2

−2

−1

10

10

−1

10

X/h

10

X/h

Figure 13.1. The footprint function of concentration c (z01 D 1 cm, z02 D 5 cm (left panel), and z01 D 1 cm, z02 D 25 cm (right panel)), vs. the dimensionless upwind distance X= h, for different values of xd .

C

C

xd = 100 m

180

160

x = 20 m

160

d

140

xd = 50 m

xd = 50 m

140 120

xd = 20 m

120

x = 100 m

100

xd = 200 m

homogeneous z = 5 cm 0

d

100

x = 200 m

80

d

80 60

60

homogeneous, z = 25 cm 0

40

40 20

20

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

X/h

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

X/h

Figure 13.2. The cumulative footprint functions of concentration C , versus the dimensionless upwind distance X= h, for different values of xd . The roughness change is the same as in Figure 13.1.

269

Section 13.3 Results

q

q

35

homogeneous z = 5 cm

20

30

0

homogeneous z = 25 cm 0

25

15

20

xd = 100 m

10

xd = 20 m

15

xd = 50 m

10

x = 50 m

5

d

xd = 100 m

5

0

xd = 20 m

0 −2

−2

−1

10

−1

10

10

10

X/h

X/h

Figure 13.3. The footprint function of flux q (z01 D 1 cm, z02 D 5 cm (left panel), and z01 D 1 cm, z02 D 25 cm (right panel)), vs. the dimensionless upwind distance X= h, for different values of xd .

Q

Q 1.4

x = 20 m

x = 20 m d

xd = 100 m

x = 50 m

1.2

xd = 50 m

d

1.2

d

xd = 100 m

1

1

x = 200 m

0.8

0.8

xd = 200 m

homogeneous z0 = 5 cm

d

homogeneous z = 25 cm 0

0.6

0.6

0.4

0.4

0.2

0.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

X/h

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

X/h

Figure 13.4. The cumulative footprint functions of flux Q, vs. the dimensionless upwind distance X= h, for different values of xd . The roughness change is the same as in Figure 13.3.

270

Q

n

Chapter 13 Lagrangian footprint calculations

Qn

x = 200 m d

1

1

0.9

0.9

x = 50 m d

0.7

d

0.6

0

x = 50 m d

0.5

0.5

xd = 20 m

0.4

xd = 20 m

0.4

0.3

0.3 0.2

0.2

0.1

0.1

0

homogeneous z = 25 cm

xd = 200 m

0.7

homogeneous z = 5 cm 0

0.6

xd = 100 m

0.8

x = 100 m

0.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0

0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

X/h

0.9

X/h

Figure 13.5. The normalized cumulative footprint functions of flux Q, vs. the dimensionless upwind distance X= h, for different values of xd . The roughness change is the same as in Figure 13.3.

c

c

700

1000

xd = 50 m

900 600

xd = 50 m 500

xd = 200 m

800 700

xd = 100 m

x = 20 m d

600 400

xd = 20 m

x = 200 m

500

300

d

400

xd = 100 m

300

200

−3

10

−2

10

homogeneous z = 1 cm 0

200

homogeneous z0 = 1 cm

100

100 −1

−3

10

X/h

10

−2

10

−1

10

X/h

Figure 13.6. The footprint function of concentration c (z01 D 5 cm, z02 D 1 cm (left panel), and z01 D 25 cm, z02 D 1 cm (right panel)), vs. the dimensionless upwind distance X= h, for different values of xd .

271

Section 13.3 Results

C

C homogeneous z = 1 cm 0

160

homogeneous z = 1 cm 0

160

140

xd = 200 m

140

xd = 200 m 120

120

x = 100 m

xd = 100 m

100

d

100

xd = 50 m 80

xd = 50 m

80

xd = 20 m

60

40

40

20

20

0

0.1

0.2

0.3

xd = 20 m

60

0.4

0.5

0.6

0.7

0.8

0.9

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

X/h

X/h

Figure 13.7. The cumulative footprint functions of concentration C , vs. the dimensionless upwind distance X= h, for different values of xd . The roughness change is the same as in Figure 13.6.

q

q

14

12

xd = 100 m

14

xd = 200 m

12

10

10

8

8

xd = 100 m

x = 200 m xd = 20 m

6

d

6

x = 20 m 4

homogeneous z = 1 cm 0

2

2

xd = 50 m 0

xd = 50 m

d

4

homogeneous z = 1 cm 0

0 −2

10

−2

−1

10

10

X/h

−1

10

X/h

Figure 13.8. The footprint function of flux q (z01 D 5 cm, z02 D 1 cm (left panel), and z01 D 25 cm, z02 D 1 cm (right panel)), versus the dimensionless upwind distance X= h, for different values of xd .

272 Q

Chapter 13 Lagrangian footprint calculations

Q

homogeneous z = 1 cm

1

1.1

homogeneous z = 1 cm

1

0

0

0.9

x = 100 m

0.8

0.8

d

x = 50 m

x = 200 m

x = 100 m

x = 200 m

d

0.6

0.6

xd = 20 m

x = 50 m d

0.5

0.4

d

d

0.7

d

0.4

x = 20 m d

0.3 0.2

0.2 0.1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

X/h

0.9

X/h

Figure 13.9. The cumulative footprint functions of flux Q, vs. the dimensionless upwind distance X= h, for different values of xd . The roughness change is the same as in Figure 13 8.

Q

n 1

Qn

xd = 100 m

xd = 20 m

0.9

xd = 200 m

0.8

x = 50 m d

xd = 200 m

homogeneous z = 1 cm 0

0.7

homogeneous z0 = 1 cm

0.6

d

xd = 50 m

1

0.8

x = 20 m

1.1

x = 100 m d

0.6 0.5

0.4

0.4 0.3

0.2

0.2 0.1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

X/h

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

X/h

Figure 13.10. The normalized cumulative footprint functions of flux Q, vs. the dimensionless upwind distance X= h, for different values of xd . The roughness change is the same as in Figure 13.8.

273

Section 13.3 Results

c

q inhomogeneous

35 1200

homogeneous z = 25 cm 0

30

1000 25 800 20

inhomogeneous

homogeneous z = 1 cm 0

600

15

homogeneous z = 25 cm 0

homogeneous z = 1 cm 0

10

400

5 200 0 −2

−1

10

−2

10

−1

10

10

X/h

X/h

Figure 13.11. The footprint functions of concentration c (left panel) and flux q (right panel). For comparison, three curves are shown: the homogeneous curves for z0 D 1 cm and z0 D 25 cm, and the inhomogeneous curve for the roughness change from z0 D 1 cm to z0 D 25 cm, for xd D 50 m.

c

q homogeneous z = 25 cm

1200

0

20

homogeneous z = 25 cm 0

1000

15 800

homogeneous z = 1 cm

inhomogeneous 600

0

homogeneous z = 1 cm

inhomogeneous

0

10

400 5 200

0 −2

10

−1

−2

10

10

X/h

−1

10

X/h

Figure 13.12. The same as in Figure 13.11, but for the rough-to-smooth case: the roughness change from z0 D 25 cm to z0 D 1 cm.

274

Chapter 13 Lagrangian footprint calculations c

c 90

1800

homogeneous z0=1 m

80

xd = 20 m

1600

x = 1000 m d

1400

70

x = 100 m d

1200 60

1000

xd = 100 m

50

800 40

600

x = 500 m

30

d

400 20

x = 500 m

homogeneous z0=1 cm

200

d

10 −2

−1

10

0

10

10

−2

−1

10

X/h

10

X/h

Figure 13.13. The footprint function of concentration for the smooth-to-rough case, the roughness height changes from z01 D 1 cm to z02 D 100 cm (left panel), and rough-to-smooth case with z01 D 1 cm, z02 D 100 cm (right panel). In all curves, zd D 20 m.

q

q

homogeneous z0=1 m

3

16

x = 20 m d

14 2.5

12 2

10

8

1.5

x = 100 m d

xd = 100 m

1

6

xd = 1000 m

xd = 500 m

4

0.5

2

x = 500 m d

0

0 −2

10

−1

10

homogeneous z =1 cm 0

0

10

−2

10

−1

10

X/h

Figure 13.14. The same as in Figure 13.13, but for the footprint function of flux q.

X/h

275

Section 13.3 Results Q

Q

homogeneous z =1 cm

1

2

xd = 100 m

xd = 500 m

0

0.9

1.8

0.8

1.6

x = 100 m d

0.7

1.4

xd = 1000 m

x = 500 m

1.2

0.6

d

x = 20 m

0.5

1

d

0.4

0.8

0.3

0.6

homogeneous z =1 m

0.4

0.2

0

0.1

0.2 0

0.5

1

1.5

0

2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

X/h

X/h

Figure 13.15. The same as in Figure 13.13, but for the cumulative footprint function of flux Q.

q

c

z =2m d

3

10

35

zd = 4 m 30

zd = 2 m 2

10

z =8m

25

d

20

1

10

15

z = 16 m d

zd = 4 m

10

0

5

10

z =8m d

zd = 16 m

0 −2

10

−1

10

−2

X/h

10

−1

10

X/h

Figure 13.16. The footprint functions of concentration c (left panel) and flux q (right panel), for xd D 100 m, and different values of the detector heights zd .

276

Chapter 13 Lagrangian footprint calculations

Q

C 1.6

160

z =8m d

zd = 4 m

1.4

140

z =8m d

1.2

zd = 4 m

120

1

100

zd = 16 m

0.8

z = 16 m

80

d

0.6

60

0.4

zd = 32 m

40

z = 32 m d

0.2

20

0 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0

0

0.2

0.4

0.6

0.8

X/h

1

1.2

1.4

1.6

1.8

2

X/h

Figure 13.17. The cumulative footprint functions of concentration C (left panel) and flux Q (right panel), for xd D 100 m, and different values of the detector heights zd .

1 Qt 0.9 Xd = 100 m

0.8

0.7

0.6

0.5

Xd = 20 m

0.4

0.3

0.2

0.1

0 −0.2

z_{01} = 25 cm, z_{02) = 1 cm

rough−to−smooth case:

0

0.2

0.4

0.6

0.8

1

1.2

X/h

Figure 13.18. The cumulative footprint function of flux Q t , the rough-to-smooth case (z01 D 25 cm, z02 D 1 cm), for xd D 20 m and xd D 100 m, and for the detector height zd D 2 m.

13.4 Discussion and conclusions A closure model was used to evaluate the mean flow and the Reynolds stress tensor required in the stochastic Lagrangian model we applied to calculate the footprint functions of concentration and its horizontal and vertical fluxes. This model provides the mean velocities and other characteristics of the flow over the roughness height.

277

Section 13.5 Appendices

A sensitivity analysis was made for the footprint functions under perturbation of the roughness height; two cases were considered: (i) smooth-to-rough and (ii) roughto-smooth change of the roughness height. The calculations show that the footprint function of concentration is more sensitive than that of the vertical flux. It iwa concluded that the footprint and cumulative footprint functions of concentration for horizontally homogeneous surface, widely used in for estimating the sufficient fetch for measurements, can be seriously biased in many cases of practical importance. The calculations show that the footprint area based on the cumulative concentrations, if estimated through the homogeneous case, can be essentially under- or overestimated, compared to the true inhomogeneous case. For instance, in the case when the detector is placed at xd D 100 m from the roughness jump (from z01 D 1 cm to z02 D 25 cm), at the height zd D 2 m, the fetch calculated from the homogeneous curve is about 100 m, while the inhomogeneous curve predicts the fetch of 400 m (see the right panel of Figure 13.5). In the smooth-to-rough case, the cumulative footprint function of flux for the inhomogeneous case is larger than 1, in contrast to the homogeneous case where it is always less than 1. In the rough-to-smooth case the situation is different: here the cumulative footprint function of flux can be considerably less than 1.

13.5

Appendices

13.5.1

Appendix A. Dimensionless mean-flow equations

It is convenient to work in dimensionless variables by introducing  D z= h,

 D x= h,

bg D b=G , 2

ug D u=G,

kg D k=Gh,

vg D v=G,

lg D l= h,

wg D w=G,

m D f h=G.

In the dimensionaless form, the systems of equations read @ @ug @ug @ug C wg D kg C mvg , @ @ @ @ @ @vg @vg @vg ug C wg D kg  m.ug  1/, @ @ @ @ ug

@wg @ug C D 0, @ @ and ug

 

 cbg2 @vg 2 @bg @bg @ @bg @ug 2 C , C wg D ˛b kg C kg  @ @ @ @ @ @ kg  q 1 h 1 lg D , kg D Ck lg bg . C  l0

278

Chapter 13 Lagrangian footprint calculations

with the boundary conditions: ug D 0,

vg D 0,

wg D 0,

ug D 1,

vg D 0,

at

 D 1,

at  D z0 = h, x  0,

@vg @ug D D 0 at  D 1, x > 0, @ @ @bg D 0 at  D z0 = h, and bg D 0 at @ lg D 

 D 1,

at  D z0 = h.

In the stochastic Lagrangian models, the following statistical characteristics of the N We extract these flow are used: the tensor hu0˛ u0ˇ i, and the energy dissipation rate ". 2 functions from the following closure assumptions: "N D cb =k, and 11 D h.u01 /2 i D b=Cu , @uN E 2 , 22 D h.u02 /2 i D b=Cv , 12 D hu01 u02 i D k @x1  @uN E 1 @uN E 3 0 0 13 D hu1 u3 i D k C , @x3 @x1 @uN E 2 , 23 D hu02 u03 i D k @x3 33 D h.u03 /2 i D b=Cw . where Cu D

2 bu2 C bv2 C bw , 2 2 bu

Cv D

2 bu2 C bv2 C bw , 2 2 bv

Cw D

2 bu2 C bv2 C bw , 2 2 bw

where bu , bv , bw are the universal constants in the relations hu02 i D bu2 u2 , hv 02 i D 2 u2 which are true in the surface layer with a constant shear bv2 u2 , hw 02 i D bw  0 0 2 hu w i D u . In our calculations we took bu D 2.5, bv D 2., bw D 1.25 (e. g., see [88]). All the parameters Ck , c, Cu , Cv , Cw , et c. were chosen to fit the theory of the surface layer with neutral stratification.

13.5.2 Appendix B. Lagrangian stochastic trajectory model The main input function of the Lagrangian stochastic models is the Eulerian p. d. f. which is in our case assumed to be Gaussian: ² ³ 1 pE .u; x/ D .2/3=2 .det  /1=2 exp  .ui  uN E i /ij .uj  uN Ej / . 2

279

Section 13.5 Appendices

Here ij are the elements of a matrix ƒ which is the inverse to the matrix  defined by the entries ij D h.uE i  uN E i /.uEj  uN Ej /i, i. e., ik kj D ıij , or in matrix form, ƒ D I , I being the identity matrix. The expressions for the entries of the matrix  are given in Appendix A. Forward Lagrangian trajectories In the forward trajectory model, the governing equations are dxi D ui dt , dui D ai .x, u, t / dt C where

p C0 "N d Wi .t /,



C0 "N 1 @ij @uN E i ai D  C ik .uk  uN E k / C uN Ej 2 @xj 2 @xj

 @uN E i im im @km @j m C C .uj  uN Ej /.uk  uN E k /, uN E k .uj  uN Ej / C @xj 2 @xk 2 @xj Backward Lagrangian trajectories The backward trajectories are defined by d xO D uO ds. d uO i D aO i ds C

p

C0 "N d Wi .s/,

s < t,

where @ ln pE D ai C C0 "Nij .uj  uN Ej / aO i D ai  C0 "N @ui  C0 "N 1 @ij @uN E i D C ik .uk  uN E k / C uN Ej 2 @xj 2 @xj

 @uN E i im im @km @j m C C .uj  uN Ej /.uk  uN E k /. uN E k .uj  uN Ej / C @xj 2 @xk 2 @xj

Chapter 14

Stochastic flow simulation in 3D porous media

Stochastic models and Monte Carlo algorithms for simulation of flow through porous media beyond the small hydraulic conductivity fluctuation assumptions are presented following our papers [98, 99, 196]. The hydraulic conductivity is modeled as an isotropic random field with a log-normal distribution and prescribed correlation or spectral functions. It is sampled by a Monte Carlo method based on a randomized spectral representation. The Darcy and continuity equations with the random hydraulic conductivity are solved numerically, using the successive overrelaxation method in order to extract statistical characteristics of the flow. Hybrid averaging is used: we combine spatial and ensemble averaging to get an efficient numerical procedure. We provide some conceptual and numerical comparisons of various stochastic simulation techniques and focus on the prediction of applicability of the randomized spectral models derived under the assumption of small hydraulic conductivity fluctuations.

14.1 Introduction The porosity study has received renewed attention in recent years. Motivations for studying porosity came from many applied fields, in particular, from material science, biomedicine, geology, environment, etc. Simulating flows in natural porous media such as soils, aquifers, oil and gas reservoirs is drastically complicated by the extreme heterogeneities and with insufficient data characterizing the medium. Generally the porous media is characterized by large irregularities in the sizes and forms of pores. To approximate the corresponding flow equations, a huge number of nodes is required to get practically relevant results. It should be noted that a reasonable description of the hydraulic conductivity behavior by a deterministic function meets with serious difficulties. Therefore, a natural choice used in this field is the statistical description of the hydraulic conductivity via a random field with a given statistical structure. Freeze [58] has analyzed the available data and has found the field of hydraulic conductivity to be well described by the random log-normal distribution. In hydrogeology this approach is often used for the flow analysis in saturated zone, or for the transport of a dissolved pollutant in a saturated aquifer [1,32,66,67,137,222,223,242]; see also the overview in the books [11, 34, 65]. Generally, when dealing with boundary value problems for PDEs with random parameters, one uses two main instruments to analyse the statistical characteristics of the

Section 14.1 Introduction

281

solution: (i) the small perturbation method (based on first- or higher order approximation) applied in the case of small parameter fluctuations and (ii) the direct numerical solution of PDEs for the given samples of the random inputs. The methods derived in the first-order approximation under small hydraulic conductivity fluctuation assumption are widely used [8, 66]. However, they have strong restrictions, and as a rule, the applicability conditions are uncertain. Therefore it is important to develop a general direct numerical method which is able to provide calculations beyond the small fluctuation assumptions; in addition, it could be used to validate the small perturbation results. On the other hand, the results obtained by the first approximation method are very useful, since they are explicit, and can therefore be used as a benchmark for testing the complicated direct numerical method. In [196] we constructed a randomized spectral model (RSM) for the simulation of a steady flow in porous media in the 3D case under small fluctuation assumptions. The method follows the following scheme: first we explicitly derive the spectral tensor of the velocity from that of the hydraulic conductivity, then we construct a Monte Carlo simulation technique for the random velocity with the derived spectral tensor. Note that a small perturbation analysis using a simulation formula inspired by [102], based on a numerical evaluation of the random field representation through the spectral measure, was also applied in Schwarze et al. [219]. In this approach, when constructing the realizations of a random field, the wave vectors are sampled in the whole space. This may cause a poor statistical representation for large wave vectors, which in turn may lead to large errors for small scale evaluations such as the mean squared separation of two particles. We suggest a different simulation technique which uses a stratified sampling of wave vectors described in [119] and further developed in [104]. Concerning the higher-order corrections of the spectral expansion method, we mention that Dagan [33] derived a second-order correction of the head covariances in the 3D case. He noted that the first-order approximation is very robust, even for a logconductivity variance equal to unity, and the second-order correction of the head variance is smaller than 10 % of the first-order approximation. Thus, for small to moderate values of f2 , it is suggested that the first-order approximation is accurate enough. Similar research on velocity covariance has been carried out in [38]. These authors explored the accuracy of the first-order approximation and reported that for f2  1, the second-order corrections to the velocity covariance are unimportant, but as f2 approaches unity they become significant. We mention a high-order perturbation approach via the Karhunen–Loève decomposition reported in [265]. In this method, the log hydraulic conductivity Y is decomposed into an infinite series on the basis of a set of noncorrelated Gaussian standard random variables. The coefficients of the series are related to eigenvalues and eigenfunctions of the covariance function of the log hydraulic conductivity. The advantage of this method is that it suggests an approximation up to the fourth order in Y . However its practical use is limited by the need for solving the complicated eigenvalue problem.

282

Chapter 14 Stochastic flow simulation in 3D porous media

In the general case, when the fluctuations are not small, the only rigorous way to tackle the problem is direct numerical simulation, which allows us to analyze flows in complex domains, although it requires large computing resources. There have been some attempts to develop direct numerical simulation for the problem of transport in porous media. In [222, 223] an analysis of one- and 2-dimensional steady groundwater flows in the bounded domain is carried out. The modeling domain has a block structure with a prescribed correlation of the hydraulic conductivity in the neighboring blocks. Thus for such a simple piecewise-constant approximation of the hydraulic conductivity, the authors [222, 223] solved the Darcy equation by a finite-difference method to get samples of the hydraulic potential. It should be noted that their method neither guaranties homogeneity of the generated fields nor a specified correlation structure. An improved version based on a direct matrix inversion method is used in [12], which, however, is still time-consuming. Another attempt to construct a model of a 3-dimensional stationary saturated flow beyond the small hydraulic conductivity fluctuations assumption was made in [1]. In this paper, the estimation of the head variance was calculated, and a comparison with the first order approximation results was carried out. However, the authors were faced with the demand fpr large computer resources: it was concluded that in oder to obtain reasonable computational results, the domain and the number of nodes should be increased up to an unrealistic level (about 106 nodes). Statistical characteristics of the velocity field were also estimated by direct numerical simulation in [120], where the authors developed a stochastic Lagrangian model for the transport in a statistically isotropic porous medium. However the accuracy in these simulations also was not high enough to make the desired definite conclusions. The applicability of the first-order approximation models for the velocity covariance in the mean flow direction in the 2-dimensional case was examined in [76]. It was found that these models give very accurate results for the longitudinal velocity covariance for the values of f up to unity. However, the transversal velocity covariance deviates from the direct numerical simulation results as f approaches unity. Chin and Wang [23] used Monte Carlo simulation for a 3-dimensional flow to investigate the accuracy of the first-order approximation, in relation to the Eulerian– Lagrangian covariance relationship. They used the turning bands algorithm due to Thompson et al. [241]. This method superimposes independent random processes constructed along the lines; this is a kind of projection method constructed from the 1-dimensional spectral representations. It also cannot be considered to be an efficient simulation method, and in [23] the authors had to restrict the calculations to a crude mesh. In this chapter we present a direct simulation model in three dimensions based on a numerical evaluation of the random realizations of the hydraulic conductivity by the successive overrelaxation method (DSM-SOR method). The samples of the hydraulic log-conductivity are constructed by a randomized spectral method. Since the

283

Section 14.2 Formulation of the problem

DSM-SOR method works for arbitrary large fluctuations, we are able to investigate the applicability of the models derived in the first-order approximation. The results extracted from the numerical simulations are also useful for the parametrization of the Lagrangian stochastic model developed in [120]. Note that both in the DSM-SOR method and in the first-order approximation approach we use the randomized spectral method to simulate the random fields with the desired spectral tensor. Hence, the construction of an efficient random field simulation method is a very important issue in this study. Interesting insights into the dynamics of transport in disordered media can be already achieved through relatively simple random models for the velocity field, such as finite superposition of Fourier modes, with each amplitude independently evolving according to an Ornstein–Uhlenbeck process [26, 220]. Instead, we will use randomized spectral methods for scalar and vector Gaussian fields described in [191] and further developed in [104]. This chapter is organized as follows. We start by formulating the stochastic boundary value problem in Section 14.2. The DSM-SOR method is described in Section 14.3. The first-order approximation and the relevant randomized spectral model are presented in Section 14.4. Section 14.5 includes calculations aimed at testing the DSMSOR method by comparing with the results obtained by the randomized spectral model under small fluctuation assumptions. The main numerical simulation results obtained by the DSM-SOR method for the general case of hydraulic conductivity fluctuations are presented in Section 14.6.

14.2

Formulation of the problem

We consider a steady flow through saturated porous formation. For a stationary 3D flow, the specific discharge is determined by the Darcy law q.x/ D .x/u.x/ D K.x/r.'.x//,

x 2 D IR3 ,

(14.1)

where q is the so-called Darcy’s velocity, or specific discharge, u is the pore velocity, p C z, p is the fluid pressure,  is the porosity, ' is the hydraulic potential ' D g z is the height, and K is the hydraulic conductivity assumed to be a homogeneous log-normal random field with a given spectral density. Thus q is a random field defined by (14.1) where ' is the solution of the following conservation of mass equation:  3 X @ @' K.x/ D 0. (14.2) @xj @xj j D1

The functions K and  are the key parameters of the flow. Experimental measurements show a high heterogeneous behavior of K in space with the following remarkable property: when considering K as a random field, its distribution is well approximated by a log-normal law (e. g., see [58]).

284

Chapter 14 Stochastic flow simulation in 3D porous media

The porosity  is also often considered in some models to be a random field. However, its variability is in the problems we tackle generally much smaller than that of K. We assume therefore .x/ D  D const. We will consider the hydraulic log-conductivity ln K D F C f as a statistically homogeneous random field with Gaussian distribution N.mf , f /. Here mf D F is the mean, and f is the standard deviation. We denote by Cff .r/ D hf .x/f .x C r/i the autocorrelation function, where r is the separation vector. Moreover we assume first that f is statistically homogeneous and isotropic with the exponential autocorrelation function Cff .r/ D f2 exp.r=If /,

(14.3)

where r D jrj, If is a given correlation length. The equation (14.2) will be solved in a finite domain, with the relevant boundary value conditions formulated in the next section.

14.3 Direct numerical simulation method: DSM-SOR In this section we present the direct simulation method based on the successive over relaxation iterative solution of the relevant PDE. For brevity, we will call it the DSMSOR method. In numerical simulations we deal with the following boundary value problem in the domain D D Œ0 : L1  Œ0 : L2  Œ0 : L3 :  3 X @ @' K.x/ D 0, @xj @xj

(14.4)

j D1

with the constant head (on the left and right bounds: x1 D 0 and x1 D L1 ) and impervious (on other bounds of the domain D) boundary conditions: 8 , x1 D 0 '.x/ D '0 ˆ ˆ < '.x/ D '0  JL1 , x1 D L1 . (14.5) ˆ ˆ : @'.x/ D 0 , x2 D 0, x2 D L2 , x3 D 0, x3 D L3 @n Here Jj D @h'i=@xj is the mean hydraulic gradient in xj -direction, J D .J1 , J2 , J3 / is taken in this chapter as a constant vector J D .J , 0, 0/, and '0 being a constant. The hydraulic log-conductivity ln K is assumed to be a Gaussian isotropic random field with the mean F D 3.4012 and the autocovariance (14.3).

Section 14.3 Direct numerical simulation method: DSM-SOR

285

To construct the solution of the equation (14.4), for a chosen sample of K.x/ and satisfying the boundary conditions (14.5) we use the following finite-difference scheme in the interior nodes [86]: 

 Ki 12 j l C KiC 12 j l C Kij  12 l C Kij C 12 l C Kij l 12 C Kij lC 12 'ij l  Ki 12 j l 'i1j l  KiC 12 j l 'iC1j l  Kij  12 l 'ij 1l  Kij C 12 l 'ij C1l  Kij  12 l 'ij l1  Kij C 12 l 'ij lC1 D 0

with the uniform grid h1 D h2 D h3 D h. The normal derivative on the boundary was approximated by simply using the firstorder approximation along the normal vector. The finite-difference scheme can be written in a matrix form A'h D .Di ag.A/  L  U / 'h D fh ,

(14.6)

where 'h is the approximating solution vector which depends on the indexation, fh is the vector in the right-hand side; Diag.A/ is the diagonal matrix whose entries coincide with the diagonal elements of the matrix A; analogously, L and U are the left and right triangular matrices extracted from the matrix A. We use the successive over relaxation (SOR) iterative method for solving (14.6): 'hn D w ŒDiag.A/1 .fh C L'hn C U'hn1 C .1  w/'hn1 . In [183, 184], the influence of boundary conditions in a 2-dimensional case was investigated. It was shown that beyond 3 or 4 correlation lengths (for constant head boundary conditions, or even less for impervious boundary conditions) the influence of boundary effects on the second moment of the hydraulic potential can be neglected. The influence of the impervious boundary conditions on the head covariances in 3-dimensional flow is quite small and restricted to the neighborhood of the boundary [34, 148]. The head increments are even less sensitive to the boundary conditions. In [120], all the statistical characteristics were calculated in a domain placed five correlation lengths from the head constant or impervious boundaries of the domain D. In our calculations, four correlation lengths were enough for correct the evaluation of the velocity correlations. Thus we consider two domains: the region D D Œ0 : L1  Œ0 : L2  Œ0 : L3 , and DQ D Œ4If : L1  4If  Œ4If : L2  4If  Œ4If : L3  4If . In the region D the equations (14.1), (14.2) are solved numerically by the SOR method, Q and the desired statistical characteristics are calculated in the domain D. The hydraulic potential is chosen as '0 D 100 m. To reproduce the field K, in [120] is recommended to choose the grid size h at least not larger than If =4; in [1] this value was recommended as If =5. We have taken the grid size equal to h D If =5 and h D If =6, and the time step was t D 0.25h=hui.

286

Chapter 14 Stochastic flow simulation in 3D porous media

14.4 Randomized spectral model (RSM) In this section we present the randomized spectral model (RSM) applied along with the first-order approximation expansion under assumption of small fluctuation. So let us first describe the first-order approximation model; small random perturbation about the mean values for the potential, specific discharge and pore velocity components are assumed: ' D h'i C ' 0 D H C h,

qj D hqj i C qj0 ,

uj D huj i C uj0 ,

j D 1, 2, 3.

The autocovariance (14.3) has the spectrum Sff .k/ D If3 f2 =Π2 .1 C If2 k 2 /2 ,

(14.7)

where k D .k1 , k2 , k3 / is the wave number vector, and k D jkj. Under the assumption of small hydraulic conductivity fluctuations the spectrum of specific discharge has the form [66]   kj km kl kn 2 (14.8) ıln  2 Sff .k/. Sqj ql .k/ D KG Jm Jn ıj m  k2 k In [98] the randomized simulation approach developed in [191] is used to construct a divergenceless vector field with a given spectral tensor. We have constructed Monte Carlo simulation formulas for the specific discharge perturbation q0 , and hence the velocity perturbation u0 . We simulate i D 1, 2, : : : , N independent random fields with S.k/, and then we set u

0.N /

N

  1 X 1 0 0 E .x/ D p p ki .a/ cos.ki , x/ C ki .a/ sin.ki , x/ , N iD1 p.ki /

where Ek0 i .a/ D ki a.k/, 0ki .a/ D ki a.k/,  1=2 KG kj Jm km  Sff .k/ aj .k/ D , j D 1, 2, 3, Jj  2  k ki and ki being random variables with zero mean and unit variance, and ki , ki , ki are all sampled independently. Here k is sampled according to the density p.k/, which is generally an arbitrary density function that can be chosen from rather different arguR ments. For instance, it is recommended in [191] to use p.k/ D a2 .k/= R3 a2 .k/d k. The central limit theorem ensures, under some general assumption [106], that u0.N / .x/ converges to an ergodic Gaussian random field with the spectral tensor S.k/, as N ! 1.

287

Section 14.4 Randomized spectral model (RSM)

Assuming small perturbations, hqi D KG J (see [34]), and so the velocity is modeled as u.x/ D .KG J/= C u0 .x/. A more general randomized spectral simulation method is constructed by introducing a stratified sampling of the wave numbers (see [104, 119]). Let us present the simulation formula in its general form. Let uE .x/ D u1 .x, : : : , ul .x//T , x 2 IRd be a homogeneous vector Gaussian random field with the given spectral tensor F .k/ related to the correlation tensor B.r/ D hE u.x C rE/ uE T .x/i by Z Z B.r/ D exp¹2 i kEr ºF .k/ d k, F .k/ D exp¹2 i kEr ºB.Er / d rE. (14.9) IRd

IRd

Here ./T is used for the notation of the transpose operation. Let l X p.k/ D Fjj .k/, R

j D1

and assume that  2 D IRd p.k/ d k < 1. We will use the Holeski decomposition F .k/ D p.k/ Q.k/ Q .k/,

(14.10)

where the matrix Q is defined as a complex conjugate transposition Q D QT . We denote by Q0 and Q00 as the real and imaginare parts of the tensor Q: Q.k/ D 0 Q .k/ C i Q00 .k/. Let us denote by  D supp.p/ the support of the spectral density p.k/. We choose P a subdivision of :  D niD1 i . Let ki1 , : : : , ki n0 be a family of mutually independent identically distributed random points lying in i sampled from the p. d. f. 8 Z < p.k/ , k 2 i , i2 i2 D p.k/ d k. (14.11) fi .k/ D :0,

i else, The randomization spectral model can be written in the form n0 ° n X   i X ij Q0 .kij / cos ij  Q00 .kij / sin ij uE nn0 .x/ D p n0 iD1 j D1  ± C ij Q00 .kij / cos ij C Q0 .kij / sin ij , (14.12)

where ij D 2 kij  x, and ij , ij , i D 1, : : : , n; j D 1, : : : , n0 are mutually independent and independent of the family kij standard l-dimensional Gaussian ran.1/

.l/

dom variables (with zero mean and unity covariance matrix): ij D .ij , : : : , ij /, .1/

.l/

ij D .ij , : : : , ij /.

288

Chapter 14 Stochastic flow simulation in 3D porous media

14.5 Testing the simulation procedure In this section we test the direct numerical technique by comparing the results against the calculations obtained by RSM. Obviously RSM is a reasonable approach for small fluctuations, so we compare the results mainly for f2 D 0.01, and fix  D 1.0 and If D 1. For testing the DNS-SOR method we calculate the correlation functions Cuj ul .r/ D huj .x/ul .x C r/i by the direct numerical simulation based on the SOR iterative procedure and compare them with the results obtained by the randomized spectral model constructed in [196] for the spectrum (14.8). As concluded in [196], RSM has shown good agreement with the exact results presented through a numerical integration in the spectral representation Z Sj l .k/e i.r,k/ d k, (14.13) Cuj ul .r/ D R3

where Simpson’s rule was used to evaluate the integral (14.13). The expectations in RSM were evaluated by averaging over N D 105 samples. For evaluation of statistical characteristics of a stochastic flow one usually uses two different averaging procedures: (i) space averaging [1, 120, 242] and (ii) ensemble averaging [23, 76, 222, 223]. To use the space averaging, we have to be sure that our randomized spectral model has good ergodic properties. As reported in [104], this is the case if the number of harmonics is sufficiently large, say, more than 1,000. A compromise which seeks to avoid the problems of both types of averaging is the hybrid method, or a combined averaging: first the space averaging is taken, and then the result is averaged over n independent samples, n being not so large as in the pure ensemble averaging. In the calculations presented in Figures 14.1–14.5 of Section 14.5 we use the ensemble averaging, while in Section 14.6 we use the combined averaging. In Figure 14.1 we plot the function Cu1 u1 (left panel) and Cu2 u2 (right panel) calculated by the DSM-SOR method (solid lines) and by RSM (dashed lines); both functions are presented for the longitudinal direction r1 , J D 0.01, and the expectation is calculated as an arithmetic mean over N D 104 samples. The maximal relative difference between the results of two methods for r1 D 1 was 3 % for the curves presented in the left panel and 9 % in the right panel. The statistical error of the direct simulation results was about 7 %, and 12 %, respectively; the statistical error of RSM was less than 3 %. The statistical error in calculating an ensemble average of a random estimator  p was measured by "./ D 3 = N , where 2 is the variance of the random estimator . From Figure 14.1 it can be seen that the difference between the solid and dashed curves is small everywhere except for small values of the separated vector. This is presumably caused by the limit space resolution of the DSM-SOR method.

289

Section 14.5 Testing the simulation procedure −6

5.5

−7

x 10

7

x 10

Spectral model Direct simulation

Spectral model Direct simulation

5

6

4.5 5 4 4 Cu u (r1) 2 2

1

1

Cu u (r1)

3.5

3

2.5

3

2

2 1 1.5 0

1

0.5

0

0.5

1

1.5 r

2

2.5

3

−1

0

0.5

1

1

1.5 r

2

2.5

3

1

Figure 14.1. The auto-correlation function Cu1 u1 .r/ (left panel) and Cu2 u2 .r/ (right panel) in the longitudinal direction.

To control the space resolution, which is related to the large values of the wave number k in the log-conductivity power spectrum, we introduce a cut-off in the spectrum, so that the spectrum Sff .k/ is defined on the interval Œ0, kmax . In Figure 14.2 we plot similar results for the spectrum (14.7) defined on the interval [0,15]. The total energy of this spectrum is more than 91.5 % of the total energy of the full power spectrum on Œ0, 1/. The relative error (measured as the relative difference with the results obtained by RSM) in calculations of Cu1 u1 .0/, Cu2 u2 .0/ is decreased from 8 % (left panel) and 6.5 % (right panel) to 3 %. This indicates obviously that for spectra with rapidly decaying tails we can expect the accuracy of the direct numerical method to be higher. Indeed, in [196] we used also a log-conductivity random field with the Gaussian form of the covariance ! 2 r Cff .r/ D f2 exp  2 lf whose spectrum has also a Gaussian form: Sff .k/ D

f2 lf3  5=2

exp 

lf2 k 2 4

! ,

(14.14)

p where If D lf =2. From Figure 14.3 it is seen that the velocity field corresponding to this log-conductivity spectrum with lf D 1 is simulated more precisely than in the case of the exponential auto-correlation function (14.3). As the fluctuations of the hydraulic conductivity get smaller (f ! 0) then, analogous to the case analysed by RSM, the velocities tend to have Gaussian distributions.

290

Chapter 14 Stochastic flow simulation in 3D porous media −6

5

−7

x 10

6

x 10

Direct simulation Spectral model

Direct simulation Spectral model

4.5 5 4 4

Cu u (r1) 2 2

3

1

1

Cu u (r1)

3.5

2.5

3

2

2 1 1.5 0 1

0.5

0

0.5

1

1.5 r

2

2.5

−1

3

0

0.5

1

1

1.5 r

2

2.5

3

1

Figure 14.2. The auto-correlation functions Cu1 u1 .r/ (left panel) and Cu2 u2 .r/ (right panel) in longitudinal direction. Spectrum Sff .k/ is defined on the interval Œ0, 15.

−5

1.4

−7

x 10

20

x 10

Direct simulation Spectral model

Direct simulation Spectral model

1.2 15 1

Cu u (r2) 2 2

1

1

Cu u (r1)

10 0.8

0.6 5

0.4 0 0.2

0

0

0.5

1

1.5 r1

2

2.5

3

−5

0

0.5

1

1.5 r1

2

2.5

3

Figure 14.3. The auto-correlation functions Cu1 u1 .r/ (left panel) and Cu2 u2 .r/ (right panel) in the longitudinal direction; Gaussian spectrum Sff .k/.

In Figures 14.4–14.5 the probability density functions of the longitudinal and transversal velocities are shown for two different (small and large) values of f . As predicted, for f D 0.01 the modeled densities (solid lines) are well approximated by Gaussian densities (dashed lines) with the mean hu1 i D KG J and u1 D 2.1991392 E  03, u2 D 7.7482074E  04 (left panels). For large variance values (f D 1) the velocity distributions are strongly non-Gaussian (right panels).

291

Section 14.5 Testing the simulation procedure 200

2.5 Modelled p(u1) Gaussian p(u1)

180

Modelled p(u1) Gaussian p(u1)

160

2

140

1.5 p(u1)

1

p(u )

120

100

80

1

60

40

0.5

20

0 0.29

0.292

0.294

0.296

0.298

0.3 u1

0.302

0.304

0.306

0.308

0 −1

0.31

−0.5

0

0.5 u

1

1.5

2

1

Figure 14.4. The probability density of the longitudinal velocity p.u1 / for f D 0.01 (left panel) and f D 1 (right panel).

600

8 Modelled p(u2) Gaussian p(u2)

Modelled p(u2) Gaussian p(u )

7

2

500

6 400

2

p(u )

p(u2)

5

300

4

3 200

2 100

1

0 −4

−3

−2

−1

0 u

2

1

2

3

4 −3

x 10

0 −0.5

−0.4

−0.3

−0.2

−0.1

0 u

0.1

0.2

0.3

0.4

0.5

2

Figure 14.5. The probability density of the transversal velocity p.u2 / for f D 0.01 (left panel) and f D 1 (right panel).

Note that the cross-correlation functions Cu1 u2 .r/, Cu2 u3 .r/ are identically equal to zero for arbitrary longitudinal and transversal separated vectors. This property was well confirmed by our calculations.

292

Chapter 14 Stochastic flow simulation in 3D porous media

14.6 Evaluation of Eulerian and Lagrangian statistical characteristics by the DNS-SOR method In this section we present the results of direct numerical simulation for some Eulerian and Lagrangian statistical characteristics of the flow. The key results are presented in Figure 14.7 where the Eulerian velocity autocorrelation functions are shown for different intensity fluctuations of the hydraulic conductivity, compared against the results obtained by the randomized spectral method under small fluctuation assumptions. From these curves one easily extracts the region of applicability of the small perturbation approach and RSM. Another important result is presented in Figure 14.12: here we show the behavior of the mean square separation of two particles. This complicated 2-particle Lagrangian statistical characteristic plays a crucial role in the turbulent diffusion study [145]. In the Kolmogorov inertial subrange the behavior of the mean square separation is described by the well-known Richardson’s law, which predicts a cubic dependence on time. In our case, we cannot extract a universal structure for the function 2 .t /, not depending on the initial separation. However, it has shown an interesting subdiffusion behavior in transverse direction and a superdiffusion picture in longitudinal direction. In the two subsections which follow, we calculate the expectations by the hybrid averaging: we combine spatial and ensemble averaging to get efficient numerical procedure. The Eulerian statistical characteristics were first calculated by spatial averaging over 213 nodes, and then by averaging over 300 independent samples of random velocities. The Lagrangian statistical characteristics were calculated by averaging over 252 trajectories per one sample of the random velocity, with subsequent averaging over 400 realizations of the velocity field. In all cases, the number of harmonics in the randomized spectral method was taken as n0 D 100.

14.6.1 Eulerian statistical characteristics Here we present the results of calculations of the mean Eulerian velocity and velocity autocorrelation functions. The spectrum Sff .k/ is chosen in the form (14.7) which corresponds to the exponential decorrelation. hu1 i 1, for different In Figure 14.6 we plot the mean velocity in a normalized form K GJ values of f . This normalization is convenient, since the small perturbation method concludes that the mean velocity equals to KG J = . Note that for f D 1 our mean longitudinal velocity agrees well with that obtained in [120], being approximately 4 % larger. For values of f up to 1.5 the mean velocity was also calculated in [23], where the relative difference with the result predicted by the first-order approximation was about 15 %. This difference in our calculations was 26 %, and 22 % in [120].

293

Section 14.6 DNS-SOR method 1

10

0

−1

10

1

[ θ]/[K

G

J]−1

10

−2

10

−3

10

−4

10

−2

−1

10

Figure 14.6. The behavior of

0

10

hu1 i KG J

1

10

σf

10

 1, as a function of f .

1.4

0.16 Spectral model σf=0.3 σ =0.6 f σf=1

1.2

Spectral model σf=0.3 σ =0.6 f σ =1

0.14

f

0.12 1

J σ )2

f

0.8

/(K

u u

1

2

G

0.08

2

0.6

0.06

C

C

1

u u

/(K

G

f

J σ )2

0.1

0.04 0.4

0.02 0.2

0

0

0

0.5

1

1.5 r /I

1 f

2

2.5

3

−0.02

0

0.5

1

1.5 r /I

2

2.5

3

1 f

Figure 14.7. The dimensionless functions Cu1 u1 .r=If / (left panel) and Cu2 u2 .r=If / (right panel) in the longitudinal direction at different values f in comparison against results of spectral model.

Figure 14.7 shows how good the small perturbation method and RSM work. This can be seen by comparing the RSM results with the data obtained by DSM-SOR method. Here we plot the dimensionless functions Cu1 u1 (left panel) and Cu2 u2 (right panel) in the longitudinal direction r10 D r1 =If , for f D 0.3, 0.6 and f D 1. In the left panel: as expected, the relative difference between the RSM and DSM-SOR results

294

Chapter 14 Stochastic flow simulation in 3D porous media

is rapidly increasing with the growth of the fluctuation intensity, i. e., as f increases. So, for r10 D 1, this difference behaves like 9 %, 29 % and 84 % for f D 0.3, 0.6 and 1, respectively. As to the statistical error of these calculations, it was less than 1 % for DSM-SOR method, and 2 % for RSM. In the right panel: the relative difference between the two methods (again, for r10 D 1) is less than 6 %, 31 %, and 108 % for f D 0.3, 0.6, and 1, respectively. The statistical error: less than 2 % for DSM-SOR method and 3 % for RSM. Thus the curves shown in Figure 14.7 present a clear picture of the region where the small perturbation approach and RSM can be applied, and of how fast this approximation fails as the fluctuation intensity increases.

14.6.2 Lagrangian statistical characteristics In this section we present the results of numerical simulations for some Lagrangian statistical characteristics of the flow, f D 1 is fixed. The calculations were carried out for the exponential correlations controlled by the spectrum (14.7) and by the Gaussian correlations with the spectrum (14.14). The random Lagrangian trajectory X.t / D .X1 .t /, X2 .t /, X3 .t // starting at a point x0 is defined as a function satisfying the following stochastic equation: dX D u.X/, dt

X.0/ D x0 .

(14.15)

The displacement covariances are defined by Dj l .t / D h.Xj .t /  hXj i.t //.Xl .t /  hXl i.t //i. In what follows we deal with the normalized quantities: Dj0 l D Dj l =If2 ,

j , l D 1, 2, 3,

and dimensionless time t 0 D t U=If , where U D hu1 i, and KG D exp.F /. 0 .t 0 /= 2 (left panel), and D 0 .t 0 /= 2 (right panel) In Figure 14.8, the dispersions D11 22 f f are shown. The curves seem to follow a linear law in time from, say, t 0 D 3, which would be in accordance with the classical Taylor’s formula. To confirm this, we have extended these calculations to the times up to t 0 D 30 (see Figure 14.9). From these results we can conclude that the linear law happens to be true after the time t 0 D 5, both for the longitudinal and transverse dispersions. Important Lagrangian statistical characteristic is the Lagrangian correlation tensor of velocity:   Rj l . / D h .uj .X.t //  huj .X.t //i Œ.ul .X.t C  //  hul .X.t C  //ii, where X.t / is a Lagrangian trajectory started at the time t .

295

Section 14.6 DNS-SOR method 0.35

7

Exponential Gaussian

Exponential Gaussian

0.3

0.25

4

0.2

22

f f

/(σ I )2

5

D

D11/(σf If)2

6

3

0.15

2

0.1

1

0.05

0

0

0

0.5

1

1.5

2

2.5 t’

3

3.5

4

4.5

5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t’

70

1.4

60

1.2

50

1

40

0.8

D22/(σf If)2

D11/(σf If)

2

0 0 Figure 14.8. The dimensionless functions D11 .t 0 /=f2 and D22 .t 0 /=f2 (right panel). Small time.

30

0.6

20

0.4

10

0.2

0

0

5

10

15

20

25

0

30

0

5

t’

10

15 t’

20

25

30

0 0 Figure 14.9. The dimensionless functions D11 .t 0 /=f2 and D22 .t 0 /=f2 (right panel). Large time. Gaussian spectrum Sff , (14.14).

In Figure 14.10 we show the longitudinal (left panel) and transverse (right panel) Lagrangian correlation functions. Note that the transverse correlations have negative values after t 0  1, which may be connected with a trapping and which can lead to a deviation from the linear law of the transverse dispersion. To check it, we have calculated (see Figure 14.11) the integral Z Ajj . / D 0



Rjj . 0 /d  0 .

296

Chapter 14 Stochastic flow simulation in 3D porous media

0.8

0.12 Exponential Gaussian

Exponential Gaussian

0.7

0.1

0.6 0.08

f

/(σ U)2

0.06

22

0.4

R

R11/(σf U)2

0.5

0.04

0.3 0.02 0.2

0

0.1

0

0

1

2

3

4

5 t’

6

7

8

9

−0.02

10

0

1

2

3

4

5 t’

6

7

8

9

10

Figure 14.10. The dimensionless functions R11 .t 0 /=.f U /2 (left panel) and R22 .t 0 /=.f U /2 (right panel). 1.4

0.055

Exponential Gaussian

1.2

0.05 0.045 Exponential Gaussian

f

JI]

0.04 0.035

f

θ / [σ2 K

G

0.8

0.03

22

0.6

A

A11 θ/[σ2 KG J If] f

1

0.025 0.4

0.02 0.015

0.2

0.01 0

0

1

2

3

4

5 t’

6

7

8

9

10

0.005

0

1

2

3

4

5 t’

6

7

8

9

10

Figure 14.11. The dimensionless functions A11 .t 0 / =.f2 KG J If / (left panel) and A22 .t 0 / =.f2 KG J If / (right panel).

These results are in a good qualitative agreement with the results obtained by D. Chin and T. Wang in [23], although their calculations were limited by the grid size h D If =2, which is too large to provide high accuracy. Thus the D. Chin and T. Wang results underestimate the longitudinal dispersivity (left panel) to about 30 %, and the transverse dispersivity to about 15 % compared to our results.

297

Section 14.6 DNS-SOR method 60

2.5

50

40

t

ρl

ρ2

2

2

30

20

1.5

10

1

0

0

2

4

6

8

10 t’

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

t’

Figure 14.12. The mean square separation for two particles X and Y started at .0, 0, 0/ and .0, 0, 1/, respectively. In the left panel the transverse mean square separation is presented, and in the right panel the longitudinal mean square separation is shown.

Recall that Taylor’s formula relating the dispersion tensor and the Lagrangian velocity covariance Z t

Dj l .t / D 2

.t   / Rj l . / d  0

indicates that for large times the integral Ajj should not vanish (the linear dispersion behavior for large times). This is confirmed by the calculations presented in Figure 14.11. Important Lagrangian statistical characteristic is the mean square separation for two particles. Let X.t 0 / D .X1 .t 0 /, X2 .t 0 /, X3 .t 0 // and Y.t 0 / D .Y1 .t 0 /, Y2 .t 0 /, Y3 .t 0 // be the Lagrangian trajectories of two particles initially separated by a vector r0 . The mean square separation 2 .t 0 / is defined as 2 .t 0 / D h.Y1  X1 /2 C .Y2  X2 /2 C .Y3  X3 /2 i. Hence 2 .t 0 / D l2 .t 0 / C 2t .t 0 / where l2 .t 0 / D h.Y1  X1 /2 i, and l2 .t 0 / D h.Y2  X2 /2 C .Y3  X3 /2 i are the longitudinal and transverse mean square separations, respectively. We show the functions 2t .t 0 / (left panel) and l2 .t 0 / (right panel) in Figure 14.12. Clearly, for sufficiently large time t 0 the two particles move independently, so we can expect that 2 .t 0 / D 2.Dl CD t / t 0 , according to classical dispersion law where Dl and D t are the longitudinal and transverse diffusion coefficients, respectively. However for smaller times, the motion of two particles is correlated, and the dependence of l2 .t 0 / and 2t .t 0 / on t 0 is not linear. From the results presented in Figure 14.12 it can

298

Chapter 14 Stochastic flow simulation in 3D porous media

be estimated that the linear law is true approximately after t 0 D 10. For times t 0 < 10, the longitudinal dispersion is superdiffusional l2 .t 0 / D C t t 01.4 with C t D 0.85. More complicated is the structure of the transverse dispersion. There is a time interval t 0 < 2, where 2t .t 0 / seems to follow a linear behavior, and then it switches to the subdiffusional behavior in 2  t 0  10: 2t .t 0 / D 1.32 C Cl .t 0  2/0.7 ,

(14.16)

with Cl D 0.145. This agrees well with the behavior of the Lagrangian correlation functions presented in Figure 14.10, right panel. Indeed, it suggests that the first time interval of rapidly decaying correlation corresponds to the interval of linear behavior of 2t .t 0 /, i. e., the diffusion regime is reached. Then, when the negative correlation reaches its maximum (about t 0 D 2), the trapping of particles results in the change of the function 2t .t 0 / from the linear behavior to the subdiffusional regime (14.16).

14.7 Conclusions and discussion Stochastic numerical simulation technique for flow simulation through a 3D statistically isotropic porous media was developed without small perturbation assumptions. The hydraulic conductivity was modelled as an isotropic random field with a lognormal distribution and prescribed structure of the spectral functions. It was sampled by a Monte Carlo method based on a randomized spectral representation. The Darcy and continuity equations with the random hydraulic conductivity were solved numerically, using the successive over relaxation method in order to extract statistical characteristics of the flow. Hybrid averaging iwa used: we combined spatial and ensemble averaging to get an efficient numerical procedure. The method proposed enabled us to predict the applicability of the first-order approximation model derived under the assumption of small hydraulic conductivity fluctuations. Calculations of the longitudinal and transverse dispersions, the dispersivity, and the Lagrangian correlation functions have been carried out to extract the main statistical features of the flow. In particular, the calculations predict a subdiffusional behavior of the transverse dispersion and a superdiffusional behavior of the longitudinal dispersion. It should be noted that the calculations take a lot of computer time, since the stochastic PDE is solved repetitively for many independent samples of the hydraulic conductivity field with the subsequent ensemble averaging to evaluate the desired statistical characteristics of the solution. A reasonable alternative would be a spatial averaging, but to make this approach realistic, the random field must have good ergodic properties. We applied a hybrid averaging by combining the ensemble and spatial averaging which has considerably decreased the cost of the algorithm. However we have not included here a detailed analysis of the ergodic properties of the randomized spec-

Section 14.7 Conclusions and discussion

299

tral models in 3D case; preliminary estimations show that the number of harmonics should be increased simultaneously with the refinement of the mesh used in the finitedifference method.

Chapter 15

A Lagrangian stochastic model for the transport in statistically homogeneous porous media A new type of stochastic simulation models is developed for solving transport problems in saturated porous media, which is based on a generalized Langevin stochastic differential equation. A detailed derivation of the model is presented in the case when the hydraulic conductivity is assumed to be a random field with a log-normal distribution, being statistically isotropic in space. To construct a model consistent with this statistical information, we use the well-mixed condition which relates the structure of the Langevin equation and the probability density function of the Eulerian velocity field. Numerical simulations of various statistical characteristics such as the mean displacement, the displacement covariance tensor, and the Lagrangian correlation function are presented. These results are compared against the conventional direct simulation method.

15.1 Introduction It is well known that stochastic models are well developed for solving transport problems in turbulent flows such as the transport in the atmospheric boundary layer (e. g., see [115, 181]). Stochastic models were constructed for a wide class of flows, in particular, to flows through porous media. (see [34,65]). To our knowledge, in the porous media transport only one type of stochastic models was used, namely, the random displacement method (RDM) for the hydrodynamic dispersion equation. It should be stressed that RDM can be applied only if the displacement covariance tensor is known (e. g., from measurements, or numerical simulation), and cannot be applied if the functionals of interest are evaluated at times comparable to the characteristic correlation scale of the flow. In contrast, the Lagrangian stochastic models based on the tracking particles in a random velocity field extracted from numerical solution of the flow equation (for brevity, we will call this model DSM, the direct simulation method) are free of these limitations, but the computational resources required are vast. Therefore, it is quite suggestive to construct a Langevin type stochastic model which is an approximation to DSM, and is written in the form of a stochastic differential equation for the position and velocity. It is worth mentioning that this approach is widely used in atmospheric transport problems. The basis for the Langevin type approach comes from the Kolmogorov similarity theory of fully developed turbulence [145], saying that in the inertial subrange the velocity structure tensor is a linear function in time. The linearity is the necessary condition to derive a Langevin type equation to mimic the behavior

Section 15.2 Direct simulation method

301

of the real Lagrangian trajectories. Therefore, the crucial point of the present study is to find out if in the porous media, this kind of linear law can be observed. This problem is studied by the DSM in Section 15.2. Detailed derivation of the Langevin type model is given in Section 15.3. The last section deals with the numerical simulations and comparisons with the direct simulation method.

15.2

Direct simulation method

In this section we derive Eulerian and Lagrangian statistical characteristics of the random velocity field in a porous medium. We first describe the model used to obtain the samples of the random velocity field and then analyze the properties of these random fields in the light of a Langevin type model.

15.2.1

Random flow model

Space scale To define a random flow model in a porous medium, we first need to choose the space scale at which the velocity is to be studied. In hydrogeological literature a distinction is made between several space scales [30, 34]. We conventionally consider the following. (i) The microscopic scale or molecular scale where kinetics of molecules inside the pore volume plays a role. The solid walls of the pore space may be reflecting boundaries for the molecular velocities. (ii) The pore scale is the scale where the velocities are averaged over the pore volume. (iii) The macroscopic (or laboratory) scale is several orders of magnitude higher than the pore scale. It contains a sufficient number of pores to define the so-called “representative elementary volume” (REV) where macroscopic quantity like porosity, hydraulic conductivity, etc., can be defined. In a natural formation (field), the value of a parameter over the REV is assumed pointwise. (iv) The local (or formation or field) scale is the scale of a specific aquifer or field, containing one or several geological layers. (v) The regional scale may contain a system of aquifers or basins and may extend horizontally over several tens or hundreds of kilometers. In this chapter we assume that the hydrogeological properties are defined over the REV at the macroscopic scale, and we derive a Langevin type model where these properties are assumed pointwise to predict transport over the field scale. Darcy’s law In many general flow conditions, the phenomenological Darcy’s law forms the basis of the theory of flow through porous media [34]. It is a consequence of the linearity of the equations of slow viscous flow which are obtained from the Navier–Stokes equations

302

Chapter 15 Transport in porous media

by neglecting the inertial terms. For time-independent flow conditions and saturated porous media, it is written as q.r/ D .r/ u.r/ D K.r/ r .r/,

(15.1)

where q,  , u, K, and  are all macroscopic variables depending on space vector r, and  is the effective (or kinematic) porosity. This porosity takes into account the volumes of voids effectively concerned with groundwater flow (that is, for instance, without the dead-end pores and the adherence volume of the fluid to the grains). It is upper bounded by total porosity which is the volume occupied by the pores divided by the volume of the bulk medium. q is called Darcy’s velocity and represents the groundwater flow rate, i. e., the volume of water crossing a unit area of porous medium per unit time; q is a measurable quantity, whereas u is the pore velocity, i. e., the flow rate per unit area of fluid (which is equivalent to consider that only fluid is present).  is the hydraulic potential (or pressure head). It is defined by  D p=g C z, where p is fluid pressure,  volumetric mass of the fluid, g the gravitational constant, and z the height. In , the kinematic term is always neglected due to small groundwater velocities in common applications. Finally, K, the proportionality coefficient between Darcy’s velocity and the gradient of the hydraulic potential, is called hydraulic conductivity. This parameter (and also permeability) is recognized as a key parameter for groundwater flow. Several experimental techniques (mainly pumping tests, sedimentary analysis, but also seismic, geoelectrical or tracer methods) are used to intensively measure this parameter in the laboratory or in the field scale. These measurements have put in evidence the (highly) heterogeneous behavior of K in space and have suggested the use of stochastic models. Random space function Law [122] was presumably the first who used the stochastic approach in porous media and proposed, on the basis of core analysis data from a carbonate oil field reservoir a log-normal probability density function (p. d. f.) for K. Since this proposition, there is now a large body of direct evidence to support the statement that the p. d. f. for the hydraulic conductivity is log-normal [34, 58, 65, 121]. Hydraulic log conductivity Y D ln K is therefore commonly used and assumed to be distributed according to a Gaussian distribution N.mY , Y /, where mY D hY i and Y is the standard deviation. Another parameter appearing in (15.1) and considered in some models as a random field is the porosity  . However its variability is recognized as being much smaller than hydraulic conductivity in common applications. However, a small amount of data is available for the stochastic properties of this parameter. Some linear laws, obtained on a speculative basis, have been proposed [58] to relate the porosity to log-hydraulic conductivity Y , suggesting then that porosity is normally distributed.

303

Section 15.2 Direct simulation method

The flow equation For a time-independent problem with no water source/sink, the continuity equation may be written as r q.r/ D 0. (15.2) Combining this equation with Darcy’s law (15.1), we obtain the flow equation inside the flow domain D r ŒK.r/ r .r/ D 0, r 2 D (15.3) with the following boundary conditions over the outer surface S: .r/ D FD .r/,

@.r/ D FN .r/, @n

r 2 SN .

(15.4)

Here SD and SN are parts of S where the Dirichlet and Neumann boundary conditions are used, respectively. FD and FN are given functions over SD and SN . The solution .r/ of the flow equation (15.3) with the boundary conditions (15.4) determines entirely the time-independent flow problem in a saturated porous medium, because the knowledge of the hydraulic potential .r/ everywhere in D and over S yields the groundwater velocity by applying Darcy’s law (15.1). The hydraulic conductivity K in (15.3) is considered as a random field, the hydraulic potential .r/ is therefore also a random field, and the velocity is a random vector field as well. In practice, FD and FN are often chosen as simple deterministic functions, and the boundaries are taken sufficiently far from the region of interest to avoid local effects due to the boundaries [184, 185]. Both the intensive practical measurements of the hydraulic conductivity in real applications and the central role played by this function in the flow equation (15.3) are favored for constructing a random flow model in porous media from a random conductivity model. Generation of the random conductivity field is followed by the solution of the flow equation (15.3) and by the applcation of Darcy’s law. This approach is commonly used by many authors in hydrogeology [211]. In this chapter, we will focus on the following model for Y D ln K: 1. Y is Gaussian with constant mean mY D< Y > and standard deviation Y ; 2. Y is statistically homogeneous and isotropic with an exponential auto-correlation function  ˝ 0 ˛ r 0 2 , (15.5) CY Y .r/ D Y .x C r/Y .x/ D Y exp  IY where IY is a finite and given correlation length, and r D jrj.

15.2.2

Numerical simulation

The numerical calculation of several realizations of random velocity fields and the simulation of particle trajectories by the DSM follow four principal steps : (i) generation of

304

Chapter 15 Transport in porous media

a hydraulic conductivity field with a prescribed statistical structure; (ii) evaluation of the Eulerian flow field by solving (15.3) with the boundary conditions (15.4); (iii) Identification of the particle’s instantaneous position along the path line; (iv) evaluation of the statistical moments at fixed travel time or travel distance. The accuracy of the solution depends on the numerical errors related to each step [211]. Step (i) is influenced by the choice of the generator procedure and by sampling frequency of the random field; step (ii) depends on the numerical method adopted in order to solve the flow equation (15.3) and also on the discretization; in step (iii), the error is related to the particle tracking procedure, while the convergence of step (iv) depends on the dimension of the sample used in the statistical computation. In order to limit possible inaccuracy, we considered the following numerical procedure: 1. Generation of a homogeneous random hydraulic conductivity field K.r/ in a 3D domain with the characteristics  

 

Y D ln K is normal with mean hY i D 3.4012 and standard deviation Y D 1; correlation function for log-hydraulic conductivity CY Y .r/ D hY 0 .x/Y 0 .x C r/i D Y2 exp. IrY / where Y 0 D Y  hY i and IY D 1 is an isotropic correlation length; the porosity .x/ D  D 0.5 is constant; the K-field is generated by the randomized spectral formula (e. g., see [191]) with 8192 modes.

2. Solution of the groundwater flow equation for saturated conditions and stationary problem r ŒK.r/r .r/ D 0 (15.6) and boundary condition yD0  .y/ D J y, where J D r hi1y is the mean hydraulic gradient. The pore velocity u D Kr  is computed by the FORTRAN 90 code TRACE [251] which has been modified in order to write a finite element scheme for each component of the velocity vector so that there is no need to make additional finite difference approximations when evaluating derivatives. The hydraulic potential 0 is arbitrarily chosen (0 D 100 m) and the mean hydraulic gradient is fixed at J D 0.01 1y implying a mean groundwater velocity along the y-axis and oriented towards positive y-values. The following normalized quantities are considered: e uD

u KG J 

Equation (15.6) then becomes

,

e rD

r , IY

0  .y/ e D . IY J

  e eY 0 r ee r  D 0,

(15.7)

(15.8)

305

Section 15.2 Direct simulation method

with boundary conditions e  De y . We have chosen the remaining parameters of the groundwater flow problem as follows: KG J D 0.6 ! KG .D e hY i / D 30, 

then

hY i D ln 30 D 3.4012.

The numerical values of the geometric mean KG D 30 is a plausible value for hydraulic conductivity (expressed in m=day for instance) of an aquifer of moderate permeability. 3. Simulation of particle trajectories from the Eulerian velocity field. To construct a trajectory, we numerically solve the ordinary stochastic differential equation d X.t / D u.X.t // dt

(15.9)

with the initial conditions: X.t0 / D x0 and u.X.t0 // D u0 . The velocity u is the Eulerian velocity obtained from the flow equation. For simplicity, we used the Euler scheme (15.10) X.tn / D X.tn1 / C u.X.tn1 // t where t D tn  tn1 . We define three types of domains: (a) The groundwater flow domain D defined in the problem (15.6) with deterministic boundary conditions for y D 0 and y D L. Everywhere else, a deterministic “no flow boundary” is chosen, i. e., r D 0. In our test problem, the dimensions of this domain are Œ0, 50 Œ0, 70 Œ0, 50 corresponding to 50 times the unit correlation length IY perpendicular to the mean velocity and 70 times e unperturbed by deterministhis correlation length parallel to it. (b) The flow domain D tic boundary conditions. In hydrogeological literature [184, 185], it is well known that beyond 3 to 4 correlation lengths (or even less for impervious boundary conditions in 3D) from the deterministic boundaries the effects of these deterministic boundary conditions on the stochastic behavior of pressure head (hydraulic potential) and therefore velocity may be neglected. We have considered here 5 correlation lengths leading to e of dimensions Œ5, 45 Œ5, 65 Œ5, 45. (c) The flow domain  where the a domain D particle trajectories are initiated. We have empirically chosen a domain of dimensions Œ15, 35 Œ15, 55 Œ15, 35. This is a domain  with isotropically minimum 20 correlation lengths over which a spatial mean will be sufficiently close to the ensemble mean [227]. To reproduce the stochastic behavior of the K-field, the grid size hx D hy D hz is chosen in the flow problem (15.6) to be IY =4. In practice, minimum IY =3 or IY =4 is recommended for hx , hy or hz in the hydrological literature. We have then 201 281 201 D 11,352,681 nodes in the flow problem for each realization of the random K-field. In , a starting point x0 is systematically selected at every node of the e grid, that is, at 1,056,321 nodes. It is verified that all the trajectories remain inside D.

306

Chapter 15 Transport in porous media

A maximum simulation time T is chosen. The time steps t D tn  tn1 are variable and defined as

 Min.hx , hy , hz / . (15.11) t D tn  tn1 D Min T  tn1 ; 0.5 ku.tn1 /k Simulation is stopped when tn > T . 4. Evaluation of statistical moments at fixed travel time. We considered the following Lagrangian quantities: first, the Lagrangian correlation function for velocity. Two quantities are evaluated: the Lagrangian correlation function for the velocity component parallel to the mean flow uk and the Lagrangian correlation function for the velocity component perpendicular to the mean flow u? . They are defined by ˝ ˛ (15.12) Cuk uk .t / D .uk .t /  huk i/ .uk .0/  huk i/ , ˝ ˛ Cu? u? .t / D .u? .t /  hu? i/ .u? .0/  hu? i/ . (15.13) Next we consider the Lagrangian velocity structure functions, which are defined as hVi .t / Vj .t /jX.t0 / D x0 ; V .t0 / D u0 i,

i , j D 1, 2, 3,

(15.14)

where Vi .t / D Vi .X.t ; t0 , x0 //  Vi .t0 /. The quantities Vi .tn / Vj .tn /

i , j D 1, 2, 3 and for tn 2 Œ0, T 

(15.15)

are memorized in a table according to classes for the values of u0k and u0? . Taking into account the property of homogeneity of the random K-field and thus the random Eulerian velocity field, the arithmetic mean of the quantities .15.15/ over the number of starting points multiplied by the number of realizations of the random K-field is an approximation of the conditional means ˛ ˝ Vi .tn / Vj .tn /jV .t0 / D u0 , regrouped in classes for the values of u0k and u0? . The quantities introduced above will be analyzed in Section 15.2.4 in more detail. We only mention here that in order to evaluate these quantities the time step is reduced in an appropriate way each time the simulation time overshoots the fixed times chosen at the beginning of the simulations.

15.2.3 Evaluation of Eulerian characteristics Due to the symmetry of the flow problem, the Eulerian velocity .u, v, w/ is conveniently decomposed into a longitudinal velocity component uk  v parallel to the p mean flow and a perpendicular component u? D u2 C w 2 perpendicular to the mean flow.

307

Section 15.2 Direct simulation method

We are interested in the statistical properties of the Eulerian velocity field, i. e., mainly the probability density function pE .u, v, w/. We decompose this p. d. f. as follows: 1 k C pE .u, v, w/ D p .uk / pE .u? juk /, (15.16) 2  u? E k

where pE .uk / is the marginal p. d. f. of the Eulerian longitudinal velocity component C .u ju / is the conditional p. d. f. of the Eulerian transverse velocity uk  v and pE ? k component u? under the condition that the longitudinal component is given. By definition we have Z C1 Z 1 Z C1 k du dw pE .u, v, w/ D 2  u? pE du? D pE .v/. (15.17) 1

1

0

From the numerical procedure described in Section 15.2.2 above, we show in Figk ure 15.1 the histogram of the marginal p. d. f. pE .uk / obtained with about 1 million samples of velocities. In Figure 15.2 we show the histogram of the conditional p. d. f. C .u? juk / for different values of uk . Both figures are obtained for Y D 1. Statistical pE calculations give huk i D 0.73, u2k D 0.33; hu? i D 0.246 and u2? D 0.059. k

We notice that the p. d. f.’s are asymmetric with quite heavy tails. For pE .uk /, negative uk values are possible (with very small probability), meaning counter-current C velocities. For pE .u? juk /, the shape of the p. d. f.’s highly depends on the conditional value of uk : the higher the value of uk (in absolute value) is, the larger is the interval of possible values for u? . It is well known in the hydrogeological literature that no exact analytical expressions can be found for the p. d. f. of the Eulerian velocity. Numerous theoretical studies [80, 183] suggest approximations for the first and second velocity moments from Darcy’s law and the flow equation by applying perturbation methods of different order in Y2 . However these studies cannot describe the type of velocity p. d. f. in general cases. These limitations can be avoided by numerical studies, like in [211], where the authors thoroughly analyzed the 1st, 2nd and 3rd moments of the velocity p. d. f.’s without giving hints about a possible p. d. f. family that could fit their numerical results. We propose in this chapter to fit the numerical p. d. f.’s with generalized Weibull distributions. We assume that for a given Y ²

 ³ p2 u? C p4 p2 C p1 1 .u? C p4 / pE .u? juk / D p1 exp  , (15.18) p3 p3 .p1 =p2 / k pE .uk /

²

 ³ u k C q 4 q2 q2 q1 1 .uk C q4 / D q1 exp  , q3 q3 .q1 =q2 /

(15.19) k

C and pE are where p3 and q3 > 0 ; p1 , p2 , q1 and q2  1 and p4 and q4  0. pE generalized Weibull p. d. f.’s with shape parameters p1 and q1 , exponents p2 and q2 , scale parameters p3 and q3 , and shift parameters p4 and q4 .

308

Chapter 15 Transport in porous media 0.14 Sigma_Y = 1 0.12

Relative frequency

0.1

0.08

0.06

0.04

0.02

0 0

1

2 u_0 parallel

3

4

Figure 15.1. Histogram of the marginal p. d. f. pE .u0k /. 0.025 Sigma_Y = 1; u0_para = -0.045 u0_para = 0.070 u0_para = 0.300 u0_para = 0.531 u0_para = 1.107 u0_para = 2.028 u0_para = 3.065

Relative frequency

0.02

0.015

0.01

0.005

0 0

0.1

0.2

0.3

0.4

0.5

u_0 perpendicular

Figure 15.2. Histogram of the conditional p. d. f. pE .u0? ju0k /.

0.6

0.7

0.8

309

Section 15.2 Direct simulation method 0.45

Scale parameter of the Generalized Weibull

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -0.5

0

0.5

1 u0_parallel

1.5

Figure 15.3. Dependence of the conditional p. d. f. pE .u0? ju0k / on u0k .

k

C It should be noted that the choice of Weibull p. d. f.’s for pE and pE is not purely arbitrary; it is indeed a quite general p. d. f. family which includes the Gaussian and Gamma p. d. f.’s. In [211], it is shown that for Y2  1, both longitudinal and transversal velocities are distributed according to a Gaussian p. d. f.. Moreover, if the transversal components u and w are p independently normally distributed with equal variances, we can show that u? D u2 C w 2 is distributed according to the classical Weibull distribution with exponent p2 D 2 and shape parameter p1 D p2 . C , the parameters p1 to p4 depend on the value of the longitudinal velocity uk , In pE the perpendicular velocity u? being nonnegative by definition, p4 D 0 in (15.18). From Figure 15.2, we can intuitively suggest that uk is playing the role of a scale C . A sensitivity analysis applied to the values of p1 , p2 and parameter for the p. d. f. pE p3 after fitting of the numerical p. d. f.’s showed that p3 is indeed the most sensitive parameter with respect to uk . In Figure 15.3 we show a law representing the depenC for u D 0.2 dency of p3 with respect to uk obtained after fitting the p. d. f.’s pE k to 2. The relation is clearly linear in uk for uk > 0. Negative values of uk are very improbable, and we can assume the following law for the scale parameter p3 to be a function of uk :

p3 .uk / D  uk C ,

(15.20)

310

Chapter 15 Transport in porous media 1 Perpendicular component Parallel component 0.8

C_VV(t)/Sigma^2_V

0.6

0.4

0.2

0

-0.2 0

1

2

3 4 Non-dimensional time : tu/I_Y

5

6

Figure 15.4. Lagrangian correlation functions for parallel and perpendicular velocity components.

where ,  > 0 and  >  q4 . In Figure 15.3, we find, for instance,  D 0.217 and  D 0.01.

15.2.4 Evaluation of Lagrangian characteristics By the direct simulation method described in Section 15.2.2, we have computed the Lagrangian correlation function Cuk uk .t / for the component parallel to the mean flow uk and the Lagrangian correlation function Cu? u? .t / for the velocity component perpendicular to the mean flow u? defined in (15.12) and (15.13). In what follows we define u D huk i. In Figure 15.4, the normalized correlation functions are shown versus nondimensional time; i. e., Cuk uk =u2k and Cu? u? =u2? vs. ut =IY for Y D 1. The results shown in this figure are similar to those obtained by Saladin and Fiorotto [211], who thoroughly studied the numerical accuracy of the DSM. From Figure 15.4, we can obtain an estimation of the Lagrangian correlation time for the longitudinal velocity TL  8.2 in units IY =u. It is not very sensitive to the value of Y , as also found in [211]. This important characteristic time determines the validity range of the approximations introduced in the random displacement model, as seen in Section 15.4. The second quantity essential for the Langevin model developed in this chapter is the velocity structure function introduced in (15.14).

311

Section 15.2 Direct simulation method 0.035

(in units u**2)

0.144